Properties

Label 2-7406-1.1-c1-0-251
Degree $2$
Conductor $7406$
Sign $-1$
Analytic cond. $59.1372$
Root an. cond. $7.69007$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2·5-s + 2·6-s − 7-s + 8-s + 9-s + 2·10-s − 6·11-s + 2·12-s − 4·13-s − 14-s + 4·15-s + 16-s + 2·17-s + 18-s − 4·19-s + 2·20-s − 2·21-s − 6·22-s + 2·24-s − 25-s − 4·26-s − 4·27-s − 28-s − 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s + 0.577·12-s − 1.10·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.436·21-s − 1.27·22-s + 0.408·24-s − 1/5·25-s − 0.784·26-s − 0.769·27-s − 0.188·28-s − 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7406\)    =    \(2 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(59.1372\)
Root analytic conductor: \(7.69007\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7406} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7406,\ (\ :1/2),\ -1)\)

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

−7.72648145775843890513398461798, −6.94694577483831163491959339359, −5.86090834771361478740237710862, −5.53089418961033537507389671806, −4.76282355030806104343049735645, −3.73993866132901406583774552110, −3.06706246987278324198315147729, −2.18540614542783154413928963721, −2.10906272741511262570744658256, 0, 2.10906272741511262570744658256, 2.18540614542783154413928963721, 3.06706246987278324198315147729, 3.73993866132901406583774552110, 4.76282355030806104343049735645, 5.53089418961033537507389671806, 5.86090834771361478740237710862, 6.94694577483831163491959339359, 7.72648145775843890513398461798

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