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Properties

Label 2-156-156.155-c1-0-4
Degree $2$
Conductor $156$
Sign $0.704 - 0.709i$
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  + (−0.599 + 1.28i)2-s + (−1.62 − 0.599i)3-s + (−1.28 − 1.53i)4-s + 2.13·5-s + (1.74 − 1.72i)6-s + 1.52·7-s + (2.73 − 0.719i)8-s + (2.28 + 1.94i)9-s + (−1.28 + 2.73i)10-s + 2i·11-s + (1.16 + 3.26i)12-s + (1.56 + 3.24i)13-s + (−0.912 + 1.94i)14-s + (−3.47 − 1.28i)15-s + (−0.719 + 3.93i)16-s − 6.94i·17-s + ⋯
L(s)  = 1  + (−0.424 + 0.905i)2-s + (−0.938 − 0.346i)3-s + (−0.640 − 0.768i)4-s + 0.955·5-s + (0.711 − 0.702i)6-s + 0.575·7-s + (0.967 − 0.254i)8-s + (0.760 + 0.649i)9-s + (−0.405 + 0.865i)10-s + 0.603i·11-s + (0.334 + 0.942i)12-s + (0.433 + 0.901i)13-s + (−0.243 + 0.520i)14-s + (−0.896 − 0.330i)15-s + (−0.179 + 0.983i)16-s − 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766157 + 0.319144i\)
\(L(\frac12)\) \(\approx\) \(0.766157 + 0.319144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 - 1.28i)T \)
3 \( 1 + (1.62 + 0.599i)T \)
13 \( 1 + (-1.56 - 3.24i)T \)
good5 \( 1 - 2.13T + 5T^{2} \)
7 \( 1 - 1.52T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 + 6.94iT - 17T^{2} \)
19 \( 1 - 5.41T + 19T^{2} \)
23 \( 1 - 5.07T + 23T^{2} \)
29 \( 1 - 3.04iT - 29T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 - 1.82iT - 37T^{2} \)
41 \( 1 + 5.73T + 41T^{2} \)
43 \( 1 + 3.07iT - 43T^{2} \)
47 \( 1 + 5.68iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 11.1iT - 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 + 8.46T + 67T^{2} \)
71 \( 1 + 2.31iT - 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 9.06iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 16.6iT - 97T^{2} \)
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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

−13.46563974329422962560544927617, −11.96546571381977395305998367654, −10.98562781992347177165671545539, −9.833632573621536035236778993738, −9.073337554571886244134535309818, −7.43434660614050778255451729873, −6.79434576485860779596329100957, −5.50111295525735586862407166145, −4.83065044676986471512900414808, −1.54127811319344567048214592488, 1.41456909218745165134093284270, 3.50367730822606822464058208285, 5.08345357298126158955692400079, 6.05427591316615145796360844102, 7.80517589588464013669748594383, 9.041503806010439374078335574824, 10.06527754676545567946468255952, 10.79881105277939160167973365842, 11.51277431417574101984403477240, 12.73115749135979075779590973924

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