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Properties

Label 2-156-156.107-c1-0-19
Degree $2$
Conductor $156$
Sign $0.715 + 0.698i$
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  + (−0.863 + 1.11i)2-s + (1.29 − 1.15i)3-s + (−0.508 − 1.93i)4-s − 1.98i·5-s + (0.180 + 2.44i)6-s + (−2.42 − 1.40i)7-s + (2.60 + 1.10i)8-s + (0.329 − 2.98i)9-s + (2.21 + 1.71i)10-s + (−2.01 − 3.49i)11-s + (−2.89 − 1.90i)12-s + (0.235 + 3.59i)13-s + (3.66 − 1.50i)14-s + (−2.28 − 2.55i)15-s + (−3.48 + 1.96i)16-s + (6.77 + 3.91i)17-s + ⋯
L(s)  = 1  + (−0.610 + 0.791i)2-s + (0.744 − 0.667i)3-s + (−0.254 − 0.967i)4-s − 0.885i·5-s + (0.0734 + 0.997i)6-s + (−0.916 − 0.529i)7-s + (0.921 + 0.389i)8-s + (0.109 − 0.993i)9-s + (0.701 + 0.540i)10-s + (−0.607 − 1.05i)11-s + (−0.834 − 0.550i)12-s + (0.0652 + 0.997i)13-s + (0.978 − 0.402i)14-s + (−0.591 − 0.659i)15-s + (−0.870 + 0.491i)16-s + (1.64 + 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.715 + 0.698i$
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: \((2,\ 156,\ (\ :1/2),\ 0.715 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871836 - 0.355007i\)
\(L(\frac12)\) \(\approx\) \(0.871836 - 0.355007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.863 - 1.11i)T \)
3 \( 1 + (-1.29 + 1.15i)T \)
13 \( 1 + (-0.235 - 3.59i)T \)
good5 \( 1 + 1.98iT - 5T^{2} \)
7 \( 1 + (2.42 + 1.40i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.01 + 3.49i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-6.77 - 3.91i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.36 - 1.94i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.939 - 1.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.74 + 1.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.09iT - 31T^{2} \)
37 \( 1 + (2.61 + 4.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.61 - 2.08i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.06 + 1.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 + 1.24iT - 53T^{2} \)
59 \( 1 + (0.420 - 0.729i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.28 - 2.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.50 + 3.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.46 - 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.63T + 73T^{2} \)
79 \( 1 + 7.13iT - 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + (-2.80 + 1.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.30 - 10.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.12361214665391145845703567642, −12.05755605465174005060123599509, −10.38496274490758579938231643575, −9.426895606073249754582811113191, −8.548675275814677882669418602721, −7.73066624450169019878860317583, −6.61382935188700054509486459424, −5.49165069204006043898668885793, −3.57646570935115126789980698181, −1.16127848642575007388638266853, 2.73309165281621661509986386158, 3.25609385280593423308875119040, 5.09711604154582406433310889744, 7.15596979968208346457084040636, 8.012470698937679353939617171789, 9.393816251308772189174196393489, 9.977508393435866924596103745443, 10.65030259631872459011716477206, 12.00864516217337986155680014164, 12.92855726705579375526737973848

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