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Properties

Label 2-156-52.7-c1-0-9
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: $\chi_{156} (7, \cdot )$
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

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

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