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Properties

Label 2-156-39.29-c0-0-0
Degree $2$
Conductor $156$
Sign $0.711 + 0.702i$
Analytic cond. $0.0778541$
Root an. cond. $0.279023$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (−1 + 1.73i)19-s − 0.999·21-s + 25-s + 0.999·27-s − 31-s + (−1 − 1.73i)37-s + 0.999·39-s + (0.5 − 0.866i)43-s + 1.99·57-s + (0.5 − 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (−1 + 1.73i)19-s − 0.999·21-s + 25-s + 0.999·27-s − 31-s + (−1 − 1.73i)37-s + 0.999·39-s + (0.5 − 0.866i)43-s + 1.99·57-s + (0.5 − 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.0778541\)
Root analytic conductor: \(0.279023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :0),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5748867328\)
\(L(\frac12)\) \(\approx\) \(0.5748867328\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 - T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−12.87467162212949865026756229537, −12.19828706702413409649508990299, −11.07231596751651044830586872506, −10.34906402372767019976987561544, −8.761415698778100303896616633042, −7.61222166988859049414115298574, −6.82635072436709119164717233769, −5.53519951937327730915506033297, −4.12196296708099721647063286021, −1.85492671042857562741002517517, 2.86001860257039679398210868800, 4.64419865390129387614011547920, 5.45461149728077424722847653941, 6.78205128267635501739595218871, 8.426320178916794271269645742399, 9.223432634792648971063271686512, 10.42865324778073088288378406500, 11.21618870351214128473596623863, 12.16223898308493711616487105926, 13.13712256832366131695424890221

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