Properties

Label 2-182-1.1-c7-0-31
Degree $2$
Conductor $182$
Sign $-1$
Analytic cond. $56.8540$
Root an. cond. $7.54016$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 44.7·3-s + 64·4-s − 57.9·5-s − 358.·6-s + 343·7-s + 512·8-s − 181.·9-s − 463.·10-s + 435.·11-s − 2.86e3·12-s + 2.19e3·13-s + 2.74e3·14-s + 2.59e3·15-s + 4.09e3·16-s + 1.08e4·17-s − 1.44e3·18-s − 2.08e4·19-s − 3.70e3·20-s − 1.53e4·21-s + 3.48e3·22-s + 7.94e4·23-s − 2.29e4·24-s − 7.47e4·25-s + 1.75e4·26-s + 1.06e5·27-s + 2.19e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.957·3-s + 0.5·4-s − 0.207·5-s − 0.677·6-s + 0.377·7-s + 0.353·8-s − 0.0828·9-s − 0.146·10-s + 0.0986·11-s − 0.478·12-s + 0.277·13-s + 0.267·14-s + 0.198·15-s + 0.250·16-s + 0.536·17-s − 0.0585·18-s − 0.697·19-s − 0.103·20-s − 0.361·21-s + 0.0697·22-s + 1.36·23-s − 0.338·24-s − 0.957·25-s + 0.196·26-s + 1.03·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(56.8540\)
Root analytic conductor: \(7.54016\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 182,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8T \)
7 \( 1 - 343T \)
13 \( 1 - 2.19e3T \)
good3 \( 1 + 44.7T + 2.18e3T^{2} \)
5 \( 1 + 57.9T + 7.81e4T^{2} \)
11 \( 1 - 435.T + 1.94e7T^{2} \)
17 \( 1 - 1.08e4T + 4.10e8T^{2} \)
19 \( 1 + 2.08e4T + 8.93e8T^{2} \)
23 \( 1 - 7.94e4T + 3.40e9T^{2} \)
29 \( 1 + 1.54e5T + 1.72e10T^{2} \)
31 \( 1 - 3.80e4T + 2.75e10T^{2} \)
37 \( 1 - 8.38e4T + 9.49e10T^{2} \)
41 \( 1 + 6.38e5T + 1.94e11T^{2} \)
43 \( 1 - 2.49e4T + 2.71e11T^{2} \)
47 \( 1 + 1.21e6T + 5.06e11T^{2} \)
53 \( 1 - 9.58e5T + 1.17e12T^{2} \)
59 \( 1 + 1.69e6T + 2.48e12T^{2} \)
61 \( 1 + 2.11e6T + 3.14e12T^{2} \)
67 \( 1 + 3.13e6T + 6.06e12T^{2} \)
71 \( 1 + 2.83e6T + 9.09e12T^{2} \)
73 \( 1 + 1.39e6T + 1.10e13T^{2} \)
79 \( 1 + 2.77e6T + 1.92e13T^{2} \)
83 \( 1 + 5.17e6T + 2.71e13T^{2} \)
89 \( 1 + 2.81e6T + 4.42e13T^{2} \)
97 \( 1 - 6.79e6T + 8.07e13T^{2} \)
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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

−11.21784281500059457938487253738, −10.23366071610294467770621475481, −8.739155755247771320445796469218, −7.50000287398680899442480247996, −6.34803636755241643946963921658, −5.47622884192920447818158628073, −4.51415468745957709434749173566, −3.17845383401902535611923208223, −1.51898917667685947433481051873, 0, 1.51898917667685947433481051873, 3.17845383401902535611923208223, 4.51415468745957709434749173566, 5.47622884192920447818158628073, 6.34803636755241643946963921658, 7.50000287398680899442480247996, 8.739155755247771320445796469218, 10.23366071610294467770621475481, 11.21784281500059457938487253738

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