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Properties

Label 2-156-13.3-c1-0-0
Degree $2$
Conductor $156$
Sign $0.872 - 0.488i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
Motivic weight $1$
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 + 2·5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + (2.5 + 2.59i)13-s + (1 + 1.73i)15-s + (2 − 3.46i)17-s + (−2 + 3.46i)19-s − 0.999·21-s + (−3 − 5.19i)23-s − 25-s − 0.999·27-s + (−3 − 5.19i)29-s − 31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 0.894·5-s + (−0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + (0.693 + 0.720i)13-s + (0.258 + 0.447i)15-s + (0.485 − 0.840i)17-s + (−0.458 + 0.794i)19-s − 0.218·21-s + (−0.625 − 1.08i)23-s − 0.200·25-s − 0.192·27-s + (−0.557 − 0.964i)29-s − 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28992 + 0.336785i\)
\(L(\frac12)\) \(\approx\) \(1.28992 + 0.336785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2 - 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5 - 8.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 17T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-8 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.22929405833091896234376980300, −12.00842269388811516573612185027, −10.85887723372832577648194829179, −9.882345580425580149407445116653, −9.084440555132997721424777055273, −8.020467363895559666625534347247, −6.37012680269715102469464767428, −5.46363969577617734799272243899, −3.88926409756123921838097094583, −2.28208696086453269741944568752, 1.79363522398815726296353600104, 3.49047385926898507688183167604, 5.36033811416728972105281609698, 6.42820135966026690877177030048, 7.60133923080677457889925424303, 8.712898833827552243344155258594, 9.875127469912000617867265813772, 10.67066691721290065628693802890, 12.04717154581598833067683578174, 13.22413003857927670446620885563

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