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Properties

Label 2-156-52.15-c1-0-8
Degree $2$
Conductor $156$
Sign $-0.934 + 0.355i$
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  + (−1.31 − 0.521i)2-s + (−0.866 + 0.5i)3-s + (1.45 + 1.37i)4-s + (−1.90 + 1.90i)5-s + (1.39 − 0.205i)6-s + (−1.18 − 4.44i)7-s + (−1.19 − 2.56i)8-s + (0.499 − 0.866i)9-s + (3.50 − 1.51i)10-s + (−3.19 − 0.854i)11-s + (−1.94 − 0.459i)12-s + (−3.28 − 1.48i)13-s + (−0.751 + 6.45i)14-s + (0.698 − 2.60i)15-s + (0.240 + 3.99i)16-s + (1.84 + 1.06i)17-s + ⋯
L(s)  = 1  + (−0.929 − 0.368i)2-s + (−0.499 + 0.288i)3-s + (0.728 + 0.685i)4-s + (−0.853 + 0.853i)5-s + (0.571 − 0.0839i)6-s + (−0.449 − 1.67i)7-s + (−0.423 − 0.905i)8-s + (0.166 − 0.288i)9-s + (1.10 − 0.478i)10-s + (−0.962 − 0.257i)11-s + (−0.561 − 0.132i)12-s + (−0.910 − 0.413i)13-s + (−0.200 + 1.72i)14-s + (0.180 − 0.673i)15-s + (0.0601 + 0.998i)16-s + (0.447 + 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.934 + 0.355i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.934 + 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0247411 - 0.134531i\)
\(L(\frac12)\) \(\approx\) \(0.0247411 - 0.134531i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.521i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (3.28 + 1.48i)T \)
good5 \( 1 + (1.90 - 1.90i)T - 5iT^{2} \)
7 \( 1 + (1.18 + 4.44i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.19 + 0.854i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.84 - 1.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.91 - 1.04i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.32 + 4.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.13 + 5.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.89 - 5.89i)T + 31iT^{2} \)
37 \( 1 + (0.0344 - 0.128i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.31 - 0.353i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.970 + 0.970i)T - 47iT^{2} \)
53 \( 1 - 2.42T + 53T^{2} \)
59 \( 1 + (-0.519 - 1.93i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.30 + 3.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.14 + 8.00i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (14.4 - 3.86i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.48 + 5.48i)T + 73iT^{2} \)
79 \( 1 + 1.92iT - 79T^{2} \)
83 \( 1 + (-4.23 - 4.23i)T + 83iT^{2} \)
89 \( 1 + (0.809 - 3.02i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.15 + 4.30i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.22109670581218102443071327647, −11.10366492003704783318655576104, −10.38778717640820439221394751255, −10.02663367133106145491588365161, −8.095719762234769159240575405343, −7.40839343170161863204241402472, −6.45031404130835878525684679371, −4.24550847321652451828939203548, −3.07259178946905381976158226773, −0.17491964518835922431094356853, 2.34940614779902378970421778020, 4.98530053339480183428025419992, 5.88119389830087481855347432468, 7.28519431063865159808554322508, 8.263716778753657541049215043203, 9.121769746635161991613357352759, 10.15478845643343103562773034814, 11.64379603692155555910723443690, 12.07643773757944298652126754949, 12.98701529636708061906805556571

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