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Properties

Label 2-156-52.19-c1-0-4
Degree $2$
Conductor $156$
Sign $0.710 - 0.703i$
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  + (−0.737 + 1.20i)2-s + (−0.866 − 0.5i)3-s + (−0.913 − 1.77i)4-s + (0.218 − 0.218i)5-s + (1.24 − 0.676i)6-s + (3.40 + 0.913i)7-s + (2.82 + 0.209i)8-s + (0.499 + 0.866i)9-s + (0.102 + 0.424i)10-s + (1.20 + 4.50i)11-s + (−0.0989 + 1.99i)12-s + (2.32 − 2.75i)13-s + (−3.61 + 3.44i)14-s + (−0.297 + 0.0798i)15-s + (−2.33 + 3.24i)16-s + (0.140 − 0.0809i)17-s + ⋯
L(s)  = 1  + (−0.521 + 0.853i)2-s + (−0.499 − 0.288i)3-s + (−0.456 − 0.889i)4-s + (0.0975 − 0.0975i)5-s + (0.506 − 0.276i)6-s + (1.28 + 0.345i)7-s + (0.997 + 0.0741i)8-s + (0.166 + 0.288i)9-s + (0.0323 + 0.134i)10-s + (0.364 + 1.35i)11-s + (−0.0285 + 0.576i)12-s + (0.646 − 0.763i)13-s + (−0.966 + 0.919i)14-s + (−0.0769 + 0.0206i)15-s + (−0.583 + 0.812i)16-s + (0.0340 − 0.0196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.710 - 0.703i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.710 - 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789475 + 0.324624i\)
\(L(\frac12)\) \(\approx\) \(0.789475 + 0.324624i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.737 - 1.20i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-2.32 + 2.75i)T \)
good5 \( 1 + (-0.218 + 0.218i)T - 5iT^{2} \)
7 \( 1 + (-3.40 - 0.913i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.20 - 4.50i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.140 + 0.0809i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.50 + 5.62i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.41 - 2.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.09 + 5.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.13 - 2.13i)T + 31iT^{2} \)
37 \( 1 + (7.31 - 1.95i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.682 + 2.54i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.71 + 6.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.46 - 1.46i)T - 47iT^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + (4.88 + 1.30i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.88 - 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.11 - 0.565i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.15 + 4.30i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.37 - 2.37i)T + 73iT^{2} \)
79 \( 1 - 9.72iT - 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (-5.60 + 1.50i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (11.3 + 3.05i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.23132022711787905010941641776, −11.93485652334388291076080363438, −11.00847237919476864552181723875, −9.912938762106684998623576590314, −8.757226795512701926406697223198, −7.75444086449921111375187813689, −6.83638023781006470566387086136, −5.47135037893791630346278749022, −4.67734635773582152658860943380, −1.59211126526490392663175200629, 1.43072011177672054963051248049, 3.58172099942528676093878648474, 4.77002653696247651480407762046, 6.34956131304691344233879239298, 8.011942141547364178950889187116, 8.697738328728738053412037660506, 10.04837315493346100093549148841, 10.97085494606758683675664657895, 11.48936575352861215287129910553, 12.42138330650058137880528286218

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