Properties

Label 2-156-156.107-c1-0-12
Degree $2$
Conductor $156$
Sign $0.965 - 0.259i$
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  + (1.32 + 0.501i)2-s + (0.680 − 1.59i)3-s + (1.49 + 1.32i)4-s + 3.37i·5-s + (1.69 − 1.76i)6-s + (−1.77 − 1.02i)7-s + (1.31 + 2.50i)8-s + (−2.07 − 2.16i)9-s + (−1.69 + 4.45i)10-s + (−2.01 − 3.48i)11-s + (3.13 − 1.48i)12-s + (−0.274 − 3.59i)13-s + (−1.83 − 2.24i)14-s + (5.36 + 2.29i)15-s + (0.479 + 3.97i)16-s + (0.0707 + 0.0408i)17-s + ⋯
L(s)  = 1  + (0.934 + 0.354i)2-s + (0.392 − 0.919i)3-s + (0.748 + 0.663i)4-s + 1.50i·5-s + (0.693 − 0.720i)6-s + (−0.670 − 0.387i)7-s + (0.464 + 0.885i)8-s + (−0.691 − 0.722i)9-s + (−0.534 + 1.40i)10-s + (−0.607 − 1.05i)11-s + (0.903 − 0.427i)12-s + (−0.0760 − 0.997i)13-s + (−0.489 − 0.599i)14-s + (1.38 + 0.591i)15-s + (0.119 + 0.992i)16-s + (0.0171 + 0.00991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.965 - 0.259i$
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: \((2,\ 156,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88681 + 0.249221i\)
\(L(\frac12)\) \(\approx\) \(1.88681 + 0.249221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.501i)T \)
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

−13.30603421803921679982162712566, −12.26009215139471638429850549480, −11.12619698736487669858221356563, −10.30853219260760496105961334600, −8.347370236057879208417341507158, −7.39071937936279936156224249458, −6.60185352537777387660344662409, −5.71369514748606167389247241331, −3.36517890055896137247595653049, −2.82459389864337813302928959878, 2.32773210329951421162635694386, 4.09848184211306906081884443600, 4.82807877122921312342947402203, 5.91209644725088782991344235096, 7.71300313635535854462852546080, 9.326430650834707618114877533101, 9.594710969387681011972524458413, 11.02254084706742436501234591179, 12.17759376941118203373342861252, 12.86021362650748253329007625934

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