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Properties

Label 2-156-156.155-c1-0-18
Degree $2$
Conductor $156$
Sign $0.935 + 0.352i$
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  + (1.17 − 0.780i)2-s + (1.26 + 1.17i)3-s + (0.780 − 1.84i)4-s + 0.662·5-s + (2.41 + 0.399i)6-s − 3.83·7-s + (−0.516 − 2.78i)8-s + (0.219 + 2.99i)9-s + (0.780 − 0.516i)10-s + 2i·11-s + (3.16 − 1.41i)12-s + (−2.56 − 2.53i)13-s + (−4.51 + 2.99i)14-s + (0.840 + 0.780i)15-s + (−2.78 − 2.87i)16-s + 1.68i·17-s + ⋯
L(s)  = 1  + (0.833 − 0.552i)2-s + (0.732 + 0.680i)3-s + (0.390 − 0.920i)4-s + 0.296·5-s + (0.986 + 0.163i)6-s − 1.44·7-s + (−0.182 − 0.983i)8-s + (0.0730 + 0.997i)9-s + (0.246 − 0.163i)10-s + 0.603i·11-s + (0.912 − 0.408i)12-s + (−0.710 − 0.703i)13-s + (−1.20 + 0.799i)14-s + (0.216 + 0.201i)15-s + (−0.695 − 0.718i)16-s + 0.407i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.935 + 0.352i$
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: \((2,\ 156,\ (\ :1/2),\ 0.935 + 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88751 - 0.343225i\)
\(L(\frac12)\) \(\approx\) \(1.88751 - 0.343225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.780i)T \)
3 \( 1 + (-1.26 - 1.17i)T \)
13 \( 1 + (2.56 + 2.53i)T \)
good5 \( 1 - 0.662T + 5T^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 - 1.68iT - 17T^{2} \)
19 \( 1 - 2.15T + 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 + 7.66iT - 29T^{2} \)
31 \( 1 + 2.15T + 31T^{2} \)
37 \( 1 - 9.03iT - 37T^{2} \)
41 \( 1 - 6.41T + 41T^{2} \)
43 \( 1 + 3.68iT - 43T^{2} \)
47 \( 1 - 6.68iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 2.87iT - 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 5.51T + 67T^{2} \)
71 \( 1 + 14.6iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + 8.10iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 - 7.92iT - 97T^{2} \)
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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

−13.10086187505717756149531486250, −12.12945547110606894631803044071, −10.69231184918464836270404845268, −9.816458104559897312575323802761, −9.384957406679336712286646695466, −7.52561811469384136977145963314, −6.15989310183429948837944116455, −4.88846260907636377384475139780, −3.55562223578032068135005302971, −2.54549259513477512163977047733, 2.65019888847083702368395616195, 3.69014715230338435650018100564, 5.55990349214680650050630898472, 6.73650168713694075525897175686, 7.32416630234147432615533052764, 8.822173531478861230541946272475, 9.595578229268063537822930601716, 11.37281468635289580429270454052, 12.56428063744742672441316725746, 13.02611773731701426982568547508

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