Properties

Degree $2$
Conductor $37030$
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 + 5-s + 7-s + 8-s − 3·9-s + 10-s − 4·11-s − 6·13-s + 14-s + 16-s − 2·17-s − 3·18-s + 20-s − 4·22-s + 25-s − 6·26-s + 28-s + 6·29-s + 8·31-s + 32-s − 2·34-s + 35-s − 3·36-s + 10·37-s + 40-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.223·20-s − 0.852·22-s + 1/5·25-s − 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s + 0.158·40-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37030\)    =    \(2 \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{37030} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 37030,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.93247724291650, −14.61053102118222, −14.12952821570513, −13.50403181345370, −13.26720233485742, −12.38381889400830, −12.19915084996317, −11.50336128869238, −11.03189411992747, −10.41293563417345, −9.943497161467979, −9.420869790172417, −8.558865567335879, −8.086555385645039, −7.626771856001726, −6.865937886645252, −6.367934368950244, −5.569184832133281, −5.272981212714408, −4.649578310129229, −4.176854322907133, −2.959303010206992, −2.555249746420485, −2.331241819698295, −1.037069227231440, 0, 1.037069227231440, 2.331241819698295, 2.555249746420485, 2.959303010206992, 4.176854322907133, 4.649578310129229, 5.272981212714408, 5.569184832133281, 6.367934368950244, 6.865937886645252, 7.626771856001726, 8.086555385645039, 8.558865567335879, 9.420869790172417, 9.943497161467979, 10.41293563417345, 11.03189411992747, 11.50336128869238, 12.19915084996317, 12.38381889400830, 13.26720233485742, 13.50403181345370, 14.12952821570513, 14.61053102118222, 14.93247724291650

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