Properties

Label 4-3234e2-1.1-c1e2-0-7
Degree $4$
Conductor $10458756$
Sign $1$
Analytic cond. $666.859$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s + 5·16-s − 8·17-s − 6·18-s − 8·19-s − 4·22-s + 4·23-s − 8·24-s − 10·25-s + 4·27-s − 8·31-s − 6·32-s + 4·33-s + 16·34-s + 9·36-s − 4·37-s + 16·38-s − 8·41-s − 20·43-s + 6·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 1.83·19-s − 0.852·22-s + 0.834·23-s − 1.63·24-s − 2·25-s + 0.769·27-s − 1.43·31-s − 1.06·32-s + 0.696·33-s + 2.74·34-s + 3/2·36-s − 0.657·37-s + 2.59·38-s − 1.24·41-s − 3.04·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10458756\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(666.859\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3234} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10458756,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 28 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 180 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
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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

−8.546207677958704399328037653021, −8.370126173621390129695589999820, −7.70647970725627220271923738030, −7.66515287981542189023396363562, −6.89803630701357608047630337438, −6.82555087595750569363787587804, −6.39464694160007385587922893641, −6.28201946977882066944995791577, −5.29945299814021959004927834765, −5.17999560291857478239417795123, −4.30919506497167881380120618170, −4.15363615994975899704543567273, −3.48880534897610477124931807098, −3.35243553816755163813653302649, −2.38330601354905729824208190344, −2.29830846019740276361077734996, −1.61175161727465132907500839358, −1.57238200269212342711657703280, 0, 0, 1.57238200269212342711657703280, 1.61175161727465132907500839358, 2.29830846019740276361077734996, 2.38330601354905729824208190344, 3.35243553816755163813653302649, 3.48880534897610477124931807098, 4.15363615994975899704543567273, 4.30919506497167881380120618170, 5.17999560291857478239417795123, 5.29945299814021959004927834765, 6.28201946977882066944995791577, 6.39464694160007385587922893641, 6.82555087595750569363787587804, 6.89803630701357608047630337438, 7.66515287981542189023396363562, 7.70647970725627220271923738030, 8.370126173621390129695589999820, 8.546207677958704399328037653021

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