Properties

Label 2-7406-1.1-c1-0-91
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 $0$

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 + 2·11-s − 2·12-s − 4·13-s + 14-s − 4·15-s + 16-s + 6·17-s + 18-s + 2·20-s − 2·21-s + 2·22-s − 2·24-s − 25-s − 4·26-s + 4·27-s + 28-s − 2·29-s − 4·30-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 + 0.603·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.447·20-s − 0.436·21-s + 0.426·22-s − 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s − 0.371·29-s − 0.730·30-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: \(0\)
Selberg data: \((2,\ 7406,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.741084634\)
\(L(\frac12)\) \(\approx\) \(2.741084634\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−7.60821221820336577790892416467, −7.02555177218682477525777938215, −6.13473731467700445988773601277, −5.84388696943192329446894069998, −5.12768921784649198679752116202, −4.67926489077139327570246676481, −3.66298258523852573646031480507, −2.68936476020173603847106516371, −1.79608625711131815655643687833, −0.814795099828760560790460168235, 0.814795099828760560790460168235, 1.79608625711131815655643687833, 2.68936476020173603847106516371, 3.66298258523852573646031480507, 4.67926489077139327570246676481, 5.12768921784649198679752116202, 5.84388696943192329446894069998, 6.13473731467700445988773601277, 7.02555177218682477525777938215, 7.60821221820336577790892416467

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