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Properties

Label 2-156-12.11-c1-0-21
Degree $2$
Conductor $156$
Sign $0.467 + 0.884i$
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  + (0.960 − 1.03i)2-s + (1.58 − 0.687i)3-s + (−0.154 − 1.99i)4-s + 0.716i·5-s + (0.812 − 2.31i)6-s + 4.43i·7-s + (−2.21 − 1.75i)8-s + (2.05 − 2.18i)9-s + (0.743 + 0.687i)10-s − 6.03·11-s + (−1.61 − 3.06i)12-s − 13-s + (4.60 + 4.26i)14-s + (0.492 + 1.13i)15-s + (−3.95 + 0.618i)16-s − 2.94i·17-s + ⋯
L(s)  = 1  + (0.679 − 0.733i)2-s + (0.917 − 0.397i)3-s + (−0.0774 − 0.996i)4-s + 0.320i·5-s + (0.331 − 0.943i)6-s + 1.67i·7-s + (−0.784 − 0.620i)8-s + (0.684 − 0.729i)9-s + (0.235 + 0.217i)10-s − 1.81·11-s + (−0.467 − 0.884i)12-s − 0.277·13-s + (1.23 + 1.13i)14-s + (0.127 + 0.293i)15-s + (−0.987 + 0.154i)16-s − 0.713i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.467 + 0.884i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.467 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57936 - 0.951855i\)
\(L(\frac12)\) \(\approx\) \(1.57936 - 0.951855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.960 + 1.03i)T \)
3 \( 1 + (-1.58 + 0.687i)T \)
13 \( 1 + T \)
good5 \( 1 - 0.716iT - 5T^{2} \)
7 \( 1 - 4.43iT - 7T^{2} \)
11 \( 1 + 6.03T + 11T^{2} \)
17 \( 1 + 2.94iT - 17T^{2} \)
19 \( 1 - 2.16iT - 19T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 3.96iT - 31T^{2} \)
37 \( 1 - 4.10T + 37T^{2} \)
41 \( 1 + 6.07iT - 41T^{2} \)
43 \( 1 - 7.50iT - 43T^{2} \)
47 \( 1 - 5.69T + 47T^{2} \)
53 \( 1 - 5.88iT - 53T^{2} \)
59 \( 1 + 1.64T + 59T^{2} \)
61 \( 1 + 4.61T + 61T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 - 0.662T + 71T^{2} \)
73 \( 1 + 4.97T + 73T^{2} \)
79 \( 1 + 7.02iT - 79T^{2} \)
83 \( 1 + 6.68T + 83T^{2} \)
89 \( 1 - 8.94iT - 89T^{2} \)
97 \( 1 - 8.97T + 97T^{2} \)
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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

−12.73630365763841964349919495866, −12.12018516498643819460928182417, −10.87448893152527634845424028875, −9.720794381400717298732480480385, −8.826518512059402169022260013014, −7.59031517640076498515151293366, −6.03701519471094596629021970624, −4.93619221721226111405565696799, −2.96073653776463315200662963692, −2.37890225537026527212097491878, 2.94059708618874096996455615974, 4.24485580028050687151235782865, 5.12894265262279994750050021954, 7.03325382580897807570531391212, 7.75432627949254005961053312274, 8.651117713411137678286653124093, 10.12645860781462023534052498257, 10.90382172952260660914357922515, 12.84060327274708170735024336743, 13.21116036768967273904336797255

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