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Properties

Label 2-156-156.107-c1-0-18
Degree $2$
Conductor $156$
Sign $0.373 + 0.927i$
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  + (0.475 − 1.33i)2-s + (1.13 + 1.30i)3-s + (−1.54 − 1.26i)4-s − 3.40i·5-s + (2.28 − 0.890i)6-s + (2.05 + 1.18i)7-s + (−2.42 + 1.45i)8-s + (−0.421 + 2.97i)9-s + (−4.53 − 1.61i)10-s + (−1.63 − 2.82i)11-s + (−0.101 − 3.46i)12-s + (2.06 + 2.95i)13-s + (2.56 − 2.17i)14-s + (4.45 − 3.86i)15-s + (0.792 + 3.92i)16-s + (0.380 + 0.219i)17-s + ⋯
L(s)  = 1  + (0.336 − 0.941i)2-s + (0.655 + 0.755i)3-s + (−0.774 − 0.633i)4-s − 1.52i·5-s + (0.931 − 0.363i)6-s + (0.777 + 0.449i)7-s + (−0.856 + 0.516i)8-s + (−0.140 + 0.990i)9-s + (−1.43 − 0.511i)10-s + (−0.492 − 0.852i)11-s + (−0.0293 − 0.999i)12-s + (0.571 + 0.820i)13-s + (0.684 − 0.581i)14-s + (1.14 − 0.997i)15-s + (0.198 + 0.980i)16-s + (0.0922 + 0.0532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.373 + 0.927i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.373 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24396 - 0.840085i\)
\(L(\frac12)\) \(\approx\) \(1.24396 - 0.840085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.475 + 1.33i)T \)
3 \( 1 + (-1.13 - 1.30i)T \)
13 \( 1 + (-2.06 - 2.95i)T \)
good5 \( 1 + 3.40iT - 5T^{2} \)
7 \( 1 + (-2.05 - 1.18i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.63 + 2.82i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.380 - 0.219i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.98 + 1.72i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.46 - 6.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.57 - 4.37i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.323iT - 31T^{2} \)
37 \( 1 + (-1.58 - 2.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.42 - 0.823i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.845 + 0.488i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.05T + 47T^{2} \)
53 \( 1 + 5.71iT - 53T^{2} \)
59 \( 1 + (-2.43 + 4.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.60 + 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.51 + 2.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.174 + 0.302i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.87T + 73T^{2} \)
79 \( 1 + 13.0iT - 79T^{2} \)
83 \( 1 + 1.55T + 83T^{2} \)
89 \( 1 + (10.7 - 6.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.991 + 1.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.95836331732900611648270948592, −11.56806961032328397792136599609, −10.97963455357526757379240102111, −9.519879312066824547427612299631, −8.833352570954638210339630430903, −8.233127575905351246498301913319, −5.47854460420688511034854565871, −4.82084956781530930232185725119, −3.61111936259222436006657576881, −1.78055478123243742071345078991, 2.65905156881068873931951894662, 4.05438730040770897136447775192, 5.88529123377581855761678492557, 7.01497364538629421156847522655, 7.59313296947031876677463827150, 8.516862558832970349916023187880, 10.05187332829006371748330549335, 11.13895633361795930811252033644, 12.55760509323906204144019016913, 13.35647993654177119901490437925

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