Properties

Degree $2$
Conductor $129430$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 129430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129430\)    =    \(2 \cdot 5 \cdot 7 \cdot 43^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{129430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 129430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.912174345\)
\(L(\frac12)\) \(\approx\) \(1.912174345\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
43 \( 1 \)
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 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
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

−13.67637701694952, −12.84062558093450, −12.43460682265429, −11.82177928535394, −11.65446541322723, −11.09436285436015, −10.57750972524545, −9.851550106603068, −9.637815927153413, −9.208687652709070, −8.636937268924191, −8.170605651648051, −7.538516436856556, −7.259155030238261, −6.519597147048690, −6.035911439378080, −5.628661367446109, −4.931179157156950, −4.428347724927151, −3.700721741790909, −2.933327944423806, −2.458270401744897, −1.952897647614620, −1.098195664954170, −0.5171678178603550, 0.5171678178603550, 1.098195664954170, 1.952897647614620, 2.458270401744897, 2.933327944423806, 3.700721741790909, 4.428347724927151, 4.931179157156950, 5.628661367446109, 6.035911439378080, 6.519597147048690, 7.259155030238261, 7.538516436856556, 8.170605651648051, 8.636937268924191, 9.208687652709070, 9.637815927153413, 9.851550106603068, 10.57750972524545, 11.09436285436015, 11.65446541322723, 11.82177928535394, 12.43460682265429, 12.84062558093450, 13.67637701694952

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