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Properties

Label 4-1502e2-1.1-c1e2-0-0
Degree $4$
Conductor $2256004$
Sign $1$
Analytic cond. $143.844$
Root an. cond. $3.46316$
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 − 3·5-s − 3·7-s − 4·8-s − 9-s + 6·10-s + 5·11-s − 2·13-s + 6·14-s + 5·16-s − 5·17-s + 2·18-s − 9·20-s − 10·22-s + 2·23-s − 2·25-s + 4·26-s − 9·28-s − 7·29-s + 3·31-s − 6·32-s + 10·34-s + 9·35-s − 3·36-s − 6·37-s + 12·40-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.34·5-s − 1.13·7-s − 1.41·8-s − 1/3·9-s + 1.89·10-s + 1.50·11-s − 0.554·13-s + 1.60·14-s + 5/4·16-s − 1.21·17-s + 0.471·18-s − 2.01·20-s − 2.13·22-s + 0.417·23-s − 2/5·25-s + 0.784·26-s − 1.70·28-s − 1.29·29-s + 0.538·31-s − 1.06·32-s + 1.71·34-s + 1.52·35-s − 1/2·36-s − 0.986·37-s + 1.89·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2256004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2256004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2256004\)    =    \(2^{2} \cdot 751^{2}\)
Sign: $1$
Analytic conductor: \(143.844\)
Root analytic conductor: \(3.46316\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2256004,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
751$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 81 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 19 T + 195 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 21 T + 221 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 19 T + 193 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 189 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 11 T + 95 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 14 T + 182 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−9.395989098474327892838254510579, −9.004383367138190390016082362628, −8.530193652165793523028085402687, −8.220044489776743758974632952082, −7.55972717865618253367899593272, −7.37751729903364867026999518781, −7.07975139850327995635241412630, −6.60907894056025437630332669376, −6.14224600660411953496400481532, −5.91198255418395069472011724177, −5.22959451594870736493853863864, −4.38281411209294588646044870171, −3.91080485265409098629765845644, −3.89180512676746301689486629416, −2.86292212847448300460006187943, −2.80139155835003669497921042661, −1.85081584166359102246518630414, −1.19798050940049225249683149573, 0, 0, 1.19798050940049225249683149573, 1.85081584166359102246518630414, 2.80139155835003669497921042661, 2.86292212847448300460006187943, 3.89180512676746301689486629416, 3.91080485265409098629765845644, 4.38281411209294588646044870171, 5.22959451594870736493853863864, 5.91198255418395069472011724177, 6.14224600660411953496400481532, 6.60907894056025437630332669376, 7.07975139850327995635241412630, 7.37751729903364867026999518781, 7.55972717865618253367899593272, 8.220044489776743758974632952082, 8.530193652165793523028085402687, 9.004383367138190390016082362628, 9.395989098474327892838254510579

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