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Properties

Label 2-156-156.107-c1-0-4
Degree $2$
Conductor $156$
Sign $0.786 - 0.617i$
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  + (0.226 − 1.39i)2-s + (−0.680 + 1.59i)3-s + (−1.89 − 0.632i)4-s + 3.37i·5-s + (2.06 + 1.31i)6-s + (1.77 + 1.02i)7-s + (−1.31 + 2.50i)8-s + (−2.07 − 2.16i)9-s + (4.70 + 0.763i)10-s + (2.01 + 3.48i)11-s + (2.29 − 2.59i)12-s + (−0.274 − 3.59i)13-s + (1.83 − 2.24i)14-s + (−5.36 − 2.29i)15-s + (3.19 + 2.40i)16-s + (0.0707 + 0.0408i)17-s + ⋯
L(s)  = 1  + (0.160 − 0.987i)2-s + (−0.392 + 0.919i)3-s + (−0.948 − 0.316i)4-s + 1.50i·5-s + (0.844 + 0.534i)6-s + (0.670 + 0.387i)7-s + (−0.464 + 0.885i)8-s + (−0.691 − 0.722i)9-s + (1.48 + 0.241i)10-s + (0.607 + 1.05i)11-s + (0.663 − 0.748i)12-s + (−0.0760 − 0.997i)13-s + (0.489 − 0.599i)14-s + (−1.38 − 0.591i)15-s + (0.799 + 0.600i)16-s + (0.0171 + 0.00991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.786 - 0.617i$
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: \((2,\ 156,\ (\ :1/2),\ 0.786 - 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939551 + 0.325070i\)
\(L(\frac12)\) \(\approx\) \(0.939551 + 0.325070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.226 + 1.39i)T \)
3 \( 1 + (0.680 - 1.59i)T \)
13 \( 1 + (0.274 + 3.59i)T \)
good5 \( 1 - 3.37iT - 5T^{2} \)
7 \( 1 + (-1.77 - 1.02i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.01 - 3.48i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0707 - 0.0408i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.648 + 0.374i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.176 - 0.306i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.448 - 0.258i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.36iT - 31T^{2} \)
37 \( 1 + (-5.25 - 9.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.18 + 2.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 2.50i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.77T + 47T^{2} \)
53 \( 1 + 7.94iT - 53T^{2} \)
59 \( 1 + (-4.06 + 7.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.63 - 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.16 + 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.29 - 2.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 6.08iT - 79T^{2} \)
83 \( 1 + 1.15T + 83T^{2} \)
89 \( 1 + (6.78 - 3.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.60 + 13.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.79852330252943923405991185704, −11.58486973121242370925287792643, −11.16201638623466136512699257417, −10.14379046934798991326750943535, −9.560300079785779288180348975085, −8.032733760044376879424733283866, −6.34955650801718724538895213971, −5.06557562255063771117136647255, −3.81898113651879437859869688025, −2.53844890874543160835159178489, 1.10317255565134952228360242502, 4.26407351284103973433447136148, 5.30938523015757946620483788508, 6.35371324418854757311719923545, 7.58389481237695675408800600527, 8.518124687121971407775646013066, 9.175252424371957527450121478608, 11.14032552122815086730520722169, 12.18243838479181357667241068523, 12.91519946121414777358905471061

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