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Properties

Label 2-356-356.355-c0-0-3
Degree $2$
Conductor $356$
Sign $1$
Analytic cond. $0.177667$
Root an. cond. $0.421505$
Motivic weight $0$
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 − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s − 19-s − 20-s − 2·21-s − 23-s − 24-s + 27-s + 2·28-s + 30-s − 31-s + 32-s − 34-s − 2·35-s − 38-s − 40-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s − 19-s − 20-s − 2·21-s − 23-s − 24-s + 27-s + 2·28-s + 30-s − 31-s + 32-s − 34-s − 2·35-s − 38-s − 40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(356\)    =    \(2^{2} \cdot 89\)
Sign: $1$
Analytic conductor: \(0.177667\)
Root analytic conductor: \(0.421505\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{356} (355, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 356,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058354802\)
\(L(\frac12)\) \(\approx\) \(1.058354802\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
89 \( 1 - T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( ( 1 - T )^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )^{2} \)
97 \( 1 + T + T^{2} \)
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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

−11.67664387409176254961069404999, −11.12641832666094511225923344933, −10.59630604998005499570730013389, −8.486875419216600821463090171005, −7.81465077644172741318499151707, −6.72180728980827205506652916631, −5.57538027644770783049218703171, −4.72657548264463000766072728667, −4.01600802562592568972815723771, −2.00491779343604680931144269321, 2.00491779343604680931144269321, 4.01600802562592568972815723771, 4.72657548264463000766072728667, 5.57538027644770783049218703171, 6.72180728980827205506652916631, 7.81465077644172741318499151707, 8.486875419216600821463090171005, 10.59630604998005499570730013389, 11.12641832666094511225923344933, 11.67664387409176254961069404999

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