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Properties

Label 2-156-52.15-c1-0-1
Degree $2$
Conductor $156$
Sign $0.676 - 0.736i$
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  + (−1.39 − 0.205i)2-s + (0.866 − 0.5i)3-s + (1.91 + 0.575i)4-s + (−1.90 + 1.90i)5-s + (−1.31 + 0.521i)6-s + (1.18 + 4.44i)7-s + (−2.56 − 1.19i)8-s + (0.499 − 0.866i)9-s + (3.06 − 2.27i)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 + (3.33 + 2.20i)16-s + (1.84 + 1.06i)17-s + ⋯
L(s)  = 1  + (−0.989 − 0.145i)2-s + (0.499 − 0.288i)3-s + (0.957 + 0.287i)4-s + (−0.853 + 0.853i)5-s + (−0.536 + 0.212i)6-s + (0.449 + 1.67i)7-s + (−0.905 − 0.423i)8-s + (0.166 − 0.288i)9-s + (0.968 − 0.720i)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 + (0.834 + 0.551i)16-s + (0.447 + 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.676 - 0.736i$
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: \((2,\ 156,\ (\ :1/2),\ 0.676 - 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729956 + 0.320376i\)
\(L(\frac12)\) \(\approx\) \(0.729956 + 0.320376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.205i)T \)
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

−12.63084597558980632173958108566, −11.76421231401066675813881620240, −11.32917958668293748455654229309, −9.721927565100833904499170341782, −9.014979549268587216948546842856, −7.83712498514722410715568475340, −7.21822918530080087054669725637, −5.75642011223827497772956202037, −3.41218274993807300703774332950, −2.19529578837512664168623252597, 1.12187147603671430247228982726, 3.62785961920550095064913536137, 4.86854281453606017872327773385, 7.03653105350803408109735440538, 7.63543791506215246027667052223, 8.693241001582101112531561382703, 9.584126612847022106969062882934, 10.66095404803677918029903314576, 11.57623965840222753494445111178, 12.59357076558487984204590569617

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