Properties

Label 4-10037-1.1-c1e2-0-0
Degree $4$
Conductor $10037$
Sign $1$
Analytic cond. $0.639967$
Root an. cond. $0.894415$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s − 4·5-s + 4·6-s − 3·7-s + 4·8-s − 2·9-s + 8·10-s − 5·13-s + 6·14-s + 8·15-s − 4·16-s − 6·17-s + 4·18-s − 19-s + 6·21-s + 5·23-s − 8·24-s + 6·25-s + 10·26-s + 10·27-s − 6·29-s − 16·30-s + 4·31-s + 12·34-s + 12·35-s − 10·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s − 1.78·5-s + 1.63·6-s − 1.13·7-s + 1.41·8-s − 2/3·9-s + 2.52·10-s − 1.38·13-s + 1.60·14-s + 2.06·15-s − 16-s − 1.45·17-s + 0.942·18-s − 0.229·19-s + 1.30·21-s + 1.04·23-s − 1.63·24-s + 6/5·25-s + 1.96·26-s + 1.92·27-s − 1.11·29-s − 2.92·30-s + 0.718·31-s + 2.05·34-s + 2.02·35-s − 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10037 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10037 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10037\)
Sign: $1$
Analytic conductor: \(0.639967\)
Root analytic conductor: \(0.894415\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10037,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad10037$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 176 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$D_{4}$ \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 7 T + 19 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 11 T + 98 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - T - 67 T^{2} - p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 51 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T - 44 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T + 57 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 7 T + 163 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 92 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−17.1397822257, −16.7821938434, −16.3924105895, −15.7904932180, −15.2859178470, −14.9489873257, −14.1610793959, −13.4143466959, −12.9880696905, −12.3017226354, −11.7822136493, −11.5679763402, −10.8746081463, −10.4089659971, −9.85633221815, −9.08429698714, −8.75213734272, −8.30608334849, −7.46532489152, −7.04486400069, −6.39316550973, −5.34059456960, −4.74699116610, −3.96046871771, −2.91293519866, 0, 0, 2.91293519866, 3.96046871771, 4.74699116610, 5.34059456960, 6.39316550973, 7.04486400069, 7.46532489152, 8.30608334849, 8.75213734272, 9.08429698714, 9.85633221815, 10.4089659971, 10.8746081463, 11.5679763402, 11.7822136493, 12.3017226354, 12.9880696905, 13.4143466959, 14.1610793959, 14.9489873257, 15.2859178470, 15.7904932180, 16.3924105895, 16.7821938434, 17.1397822257

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