Properties

Label 2-8470-1.1-c1-0-191
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.23·3-s + 4-s + 5-s − 1.23·6-s + 7-s − 8-s − 1.47·9-s − 10-s + 1.23·12-s + 4.47·13-s − 14-s + 1.23·15-s + 16-s − 2·17-s + 1.47·18-s + 0.763·19-s + 20-s + 1.23·21-s − 2.76·23-s − 1.23·24-s + 25-s − 4.47·26-s − 5.52·27-s + 28-s − 3.23·29-s − 1.23·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.713·3-s + 0.5·4-s + 0.447·5-s − 0.504·6-s + 0.377·7-s − 0.353·8-s − 0.490·9-s − 0.316·10-s + 0.356·12-s + 1.24·13-s − 0.267·14-s + 0.319·15-s + 0.250·16-s − 0.485·17-s + 0.346·18-s + 0.175·19-s + 0.223·20-s + 0.269·21-s − 0.576·23-s − 0.252·24-s + 0.200·25-s − 0.877·26-s − 1.06·27-s + 0.188·28-s − 0.600·29-s − 0.225·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8470\)    =    \(2 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(67.6332\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8470} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 1.23T + 3T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 0.763T + 19T^{2} \)
23 \( 1 + 2.76T + 23T^{2} \)
29 \( 1 + 3.23T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 - 1.23T + 37T^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 8.47T + 47T^{2} \)
53 \( 1 + 5.23T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 5.52T + 61T^{2} \)
67 \( 1 + 6T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 0.472T + 73T^{2} \)
79 \( 1 + 0.763T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 2.94T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−7.65345397981329443272174860512, −6.92798389366552246157480133359, −5.97379576247946229508789422089, −5.70818132462652036012992251068, −4.56364495027765152557081532474, −3.61941718864520282469452912212, −3.00372788027609021912620474765, −2.00479265173189871046101222147, −1.47759404608975644265129384789, 0, 1.47759404608975644265129384789, 2.00479265173189871046101222147, 3.00372788027609021912620474765, 3.61941718864520282469452912212, 4.56364495027765152557081532474, 5.70818132462652036012992251068, 5.97379576247946229508789422089, 6.92798389366552246157480133359, 7.65345397981329443272174860512

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