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Properties

Label 2-156-52.11-c1-0-2
Degree $2$
Conductor $156$
Sign $0.0717 - 0.997i$
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  + (1.12 + 0.862i)2-s + (−0.866 + 0.5i)3-s + (0.510 + 1.93i)4-s + (0.756 + 0.756i)5-s + (−1.40 − 0.187i)6-s + (−0.936 + 0.250i)7-s + (−1.09 + 2.60i)8-s + (0.499 − 0.866i)9-s + (0.194 + 1.49i)10-s + (0.0537 − 0.200i)11-s + (−1.40 − 1.41i)12-s + (2.80 − 2.26i)13-s + (−1.26 − 0.526i)14-s + (−1.03 − 0.276i)15-s + (−3.47 + 1.97i)16-s + (2.04 + 1.17i)17-s + ⋯
L(s)  = 1  + (0.792 + 0.610i)2-s + (−0.499 + 0.288i)3-s + (0.255 + 0.966i)4-s + (0.338 + 0.338i)5-s + (−0.572 − 0.0763i)6-s + (−0.353 + 0.0947i)7-s + (−0.387 + 0.921i)8-s + (0.166 − 0.288i)9-s + (0.0616 + 0.474i)10-s + (0.0162 − 0.0604i)11-s + (−0.406 − 0.409i)12-s + (0.778 − 0.627i)13-s + (−0.338 − 0.140i)14-s + (−0.266 − 0.0714i)15-s + (−0.869 + 0.493i)16-s + (0.495 + 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0717 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0717 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.0717 - 0.997i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.0717 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08757 + 1.01218i\)
\(L(\frac12)\) \(\approx\) \(1.08757 + 1.01218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12 - 0.862i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-2.80 + 2.26i)T \)
good5 \( 1 + (-0.756 - 0.756i)T + 5iT^{2} \)
7 \( 1 + (0.936 - 0.250i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.0537 + 0.200i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.04 - 1.17i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.16 + 4.34i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.03 - 1.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.36 + 2.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.27 + 6.27i)T - 31iT^{2} \)
37 \( 1 + (8.14 + 2.18i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.42 - 5.31i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.45 + 2.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.57 - 8.57i)T + 47iT^{2} \)
53 \( 1 - 0.760T + 53T^{2} \)
59 \( 1 + (9.45 - 2.53i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.05 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.30 + 0.886i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.78 + 6.66i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.62 + 1.62i)T - 73iT^{2} \)
79 \( 1 - 4.17iT - 79T^{2} \)
83 \( 1 + (4.68 - 4.68i)T - 83iT^{2} \)
89 \( 1 + (13.3 + 3.58i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.29 + 2.22i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.28103962970375446682027364456, −12.34934862822709249725467279325, −11.31090680315174063578412655105, −10.34077275176935708530970611336, −8.984837268346319956472159975249, −7.70783109303261134578324015825, −6.41087977967702703401960943449, −5.74732869216646981440663469238, −4.37298840675451291456879625167, −2.96872402120259533074159840176, 1.56359511715928511722469943058, 3.50213385506231623614337859310, 4.95018334407504424854512291733, 6.00478331904849354119242010554, 6.98763468165145181319037843738, 8.782644667392513808831806887329, 9.996242576075408164281979180261, 10.83526851273856633793149778053, 11.95491124877221088622918421045, 12.58488167924515458073016890670

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