Properties

Label 4-2450e2-1.1-c1e2-0-26
Degree $4$
Conductor $6002500$
Sign $1$
Analytic cond. $382.724$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 4·9-s − 4·11-s + 5·16-s + 8·18-s + 8·22-s + 8·23-s + 4·29-s − 6·32-s − 12·36-s − 20·37-s − 4·43-s − 12·44-s − 16·46-s + 4·53-s − 8·58-s + 7·64-s − 24·67-s − 24·71-s + 16·72-s + 40·74-s − 8·79-s + 7·81-s + 8·86-s + 16·88-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 4/3·9-s − 1.20·11-s + 5/4·16-s + 1.88·18-s + 1.70·22-s + 1.66·23-s + 0.742·29-s − 1.06·32-s − 2·36-s − 3.28·37-s − 0.609·43-s − 1.80·44-s − 2.35·46-s + 0.549·53-s − 1.05·58-s + 7/8·64-s − 2.93·67-s − 2.84·71-s + 1.88·72-s + 4.64·74-s − 0.900·79-s + 7/9·81-s + 0.862·86-s + 1.70·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6002500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(382.724\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6002500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
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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

−8.784467759916860109698174652067, −8.515948099688136973439282906171, −8.142749811253533413828290341214, −7.61880032922231220197564201244, −7.21993130346444098396225189624, −7.17201756060828013185414295338, −6.45997921700750033789536909292, −6.21407898344471964085582248513, −5.49870623595802656801151655402, −5.48231713596203108958680423804, −4.89724914111012723742836738196, −4.50446129371836406425191915992, −3.47519687889433557163866123930, −3.29999362433018355676864180741, −2.65325475390506882403929399354, −2.58572925275752240835527945302, −1.64863925254873572140399482410, −1.22729590705946039541500637626, 0, 0, 1.22729590705946039541500637626, 1.64863925254873572140399482410, 2.58572925275752240835527945302, 2.65325475390506882403929399354, 3.29999362433018355676864180741, 3.47519687889433557163866123930, 4.50446129371836406425191915992, 4.89724914111012723742836738196, 5.48231713596203108958680423804, 5.49870623595802656801151655402, 6.21407898344471964085582248513, 6.45997921700750033789536909292, 7.17201756060828013185414295338, 7.21993130346444098396225189624, 7.61880032922231220197564201244, 8.142749811253533413828290341214, 8.515948099688136973439282906171, 8.784467759916860109698174652067

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