Properties

Label 4-350e2-1.1-c5e2-0-10
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $3151.06$
Root an. cond. $7.49228$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 3·3-s + 48·4-s + 24·6-s + 98·7-s − 256·8-s + 363·9-s + 959·11-s − 144·12-s + 393·13-s − 784·14-s + 1.28e3·16-s + 2.23e3·17-s − 2.90e3·18-s + 3.34e3·19-s − 294·21-s − 7.67e3·22-s + 450·23-s + 768·24-s − 3.14e3·26-s − 2.88e3·27-s + 4.70e3·28-s − 4.51e3·29-s − 2.03e3·31-s − 6.14e3·32-s − 2.87e3·33-s − 1.78e4·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.192·3-s + 3/2·4-s + 0.272·6-s + 0.755·7-s − 1.41·8-s + 1.49·9-s + 2.38·11-s − 0.288·12-s + 0.644·13-s − 1.06·14-s + 5/4·16-s + 1.87·17-s − 2.11·18-s + 2.12·19-s − 0.145·21-s − 3.37·22-s + 0.177·23-s + 0.272·24-s − 0.912·26-s − 0.760·27-s + 1.13·28-s − 0.996·29-s − 0.380·31-s − 1.06·32-s − 0.459·33-s − 2.64·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3151.06\)
Root analytic conductor: \(7.49228\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 122500,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.905920027\)
\(L(\frac12)\) \(\approx\) \(3.905920027\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{2} T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 - p^{2} T )^{2} \)
good3$D_{4}$ \( 1 + p T - 118 p T^{2} + p^{6} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 959 T + 544442 T^{2} - 959 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 393 T + 591692 T^{2} - 393 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 2231 T + 4015832 T^{2} - 2231 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 3342 T + 7714118 T^{2} - 3342 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 450 T - 6027314 T^{2} - 450 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 4515 T + 21490372 T^{2} + 4515 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 2036 T + 23243550 T^{2} + 2036 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 1928 T + 133674294 T^{2} - 1928 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 19318 T + 258029594 T^{2} - 19318 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 14146 T + 330672654 T^{2} - 14146 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 19745 T + 544626710 T^{2} - 19745 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 26378 T + 911827778 T^{2} + 26378 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 11104 T + 899856086 T^{2} + 11104 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 3554 T + 491608410 T^{2} - 3554 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 65208 T + 2645517686 T^{2} + 65208 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 17104 T + 3301248782 T^{2} - 17104 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 36740 T + 1141498182 T^{2} - 36740 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 72245 T + 6817044654 T^{2} - 72245 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 99676 T + 8405717894 T^{2} + 99676 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 82454 T + 5676583178 T^{2} - 82454 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 104627 T + 12436884864 T^{2} - 104627 p^{5} T^{3} + p^{10} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.94752406118895925965354428757, −10.20347647640354335138460066730, −9.916065692559037843298510306893, −9.410178445713308835027478743862, −9.022144029613022244352062829260, −8.990067507819972474415114932700, −7.73561182220425883398888906940, −7.55430450906983302458436309515, −7.55129773951470098537567022211, −6.78892904640639810453873194819, −6.06800771653844250880845935656, −5.85955447715632657061535621244, −5.07068159231805790997066602294, −4.24548235527092787267679474099, −3.69671910839095726843830022113, −3.31150454301165850009202475972, −2.11952965212645115672020278223, −1.33204880016596721414626514664, −1.16681719686071393847017741404, −0.854779240102295879013040812838, 0.854779240102295879013040812838, 1.16681719686071393847017741404, 1.33204880016596721414626514664, 2.11952965212645115672020278223, 3.31150454301165850009202475972, 3.69671910839095726843830022113, 4.24548235527092787267679474099, 5.07068159231805790997066602294, 5.85955447715632657061535621244, 6.06800771653844250880845935656, 6.78892904640639810453873194819, 7.55129773951470098537567022211, 7.55430450906983302458436309515, 7.73561182220425883398888906940, 8.990067507819972474415114932700, 9.022144029613022244352062829260, 9.410178445713308835027478743862, 9.916065692559037843298510306893, 10.20347647640354335138460066730, 10.94752406118895925965354428757

Graph of the $Z$-function along the critical line