Properties

Degree $2$
Conductor $36414$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s − 4·11-s + 6·13-s − 14-s + 16-s − 4·19-s − 2·20-s + 4·22-s + 8·23-s − 25-s − 6·26-s + 28-s − 2·29-s − 32-s − 2·35-s + 10·37-s + 4·38-s + 2·40-s − 6·41-s − 4·43-s − 4·44-s − 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.447·20-s + 0.852·22-s + 1.66·23-s − 1/5·25-s − 1.17·26-s + 0.188·28-s − 0.371·29-s − 0.176·32-s − 0.338·35-s + 1.64·37-s + 0.648·38-s + 0.316·40-s − 0.937·41-s − 0.609·43-s − 0.603·44-s − 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36414\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{36414} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 36414,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.38889270209903, −14.93300718358664, −14.18296529947457, −13.42344499132335, −13.00624948425160, −12.67042166562944, −11.67958036896983, −11.41432738015213, −10.88423560422103, −10.58544142627103, −9.918348948402142, −9.032094503369328, −8.766086075477681, −8.050921922415745, −7.845681815955103, −7.228121871677442, −6.425705436740661, −6.040095793652828, −5.147549892219003, −4.636029730543089, −3.803717734184318, −3.266873730069556, −2.541390322752838, −1.668673828714373, −0.8863517208018618, 0, 0.8863517208018618, 1.668673828714373, 2.541390322752838, 3.266873730069556, 3.803717734184318, 4.636029730543089, 5.147549892219003, 6.040095793652828, 6.425705436740661, 7.228121871677442, 7.845681815955103, 8.050921922415745, 8.766086075477681, 9.032094503369328, 9.918348948402142, 10.58544142627103, 10.88423560422103, 11.41432738015213, 11.67958036896983, 12.67042166562944, 13.00624948425160, 13.42344499132335, 14.18296529947457, 14.93300718358664, 15.38889270209903

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