Properties

Label 2-156-156.35-c1-0-21
Degree $2$
Conductor $156$
Sign $-0.215 + 0.976i$
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  + (0.268 − 1.38i)2-s + (1.63 − 0.577i)3-s + (−1.85 − 0.744i)4-s − 1.45i·5-s + (−0.364 − 2.42i)6-s + (−1.17 + 0.680i)7-s + (−1.53 + 2.37i)8-s + (2.33 − 1.88i)9-s + (−2.01 − 0.388i)10-s + (−0.711 + 1.23i)11-s + (−3.46 − 0.142i)12-s + (3.27 + 1.50i)13-s + (0.629 + 1.82i)14-s + (−0.838 − 2.36i)15-s + (2.89 + 2.76i)16-s + (1.68 − 0.971i)17-s + ⋯
L(s)  = 1  + (0.189 − 0.981i)2-s + (0.942 − 0.333i)3-s + (−0.928 − 0.372i)4-s − 0.648i·5-s + (−0.148 − 0.988i)6-s + (−0.445 + 0.257i)7-s + (−0.541 + 0.840i)8-s + (0.777 − 0.628i)9-s + (−0.637 − 0.122i)10-s + (−0.214 + 0.371i)11-s + (−0.999 − 0.0412i)12-s + (0.908 + 0.417i)13-s + (0.168 + 0.486i)14-s + (−0.216 − 0.611i)15-s + (0.723 + 0.690i)16-s + (0.408 − 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900418 - 1.12039i\)
\(L(\frac12)\) \(\approx\) \(0.900418 - 1.12039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.268 + 1.38i)T \)
3 \( 1 + (-1.63 + 0.577i)T \)
13 \( 1 + (-3.27 - 1.50i)T \)
good5 \( 1 + 1.45iT - 5T^{2} \)
7 \( 1 + (1.17 - 0.680i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.711 - 1.23i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.68 + 0.971i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.01 - 1.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.96 - 3.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.48 + 4.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (0.577 - 1.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.52 - 2.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.0 + 5.80i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.49T + 47T^{2} \)
53 \( 1 - 9.80iT - 53T^{2} \)
59 \( 1 + (-2.78 - 4.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.92 + 8.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.9 + 6.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.89 + 10.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 7.21iT - 79T^{2} \)
83 \( 1 - 0.869T + 83T^{2} \)
89 \( 1 + (-4.70 - 2.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.22 + 9.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.71885417922517752177074331086, −11.95794943641513623951344598165, −10.60485455068876727800487407361, −9.413257873745090145516137735702, −8.897245628347501932084036869239, −7.72124188193622456230556213713, −5.99182256621982192276575541117, −4.40651534390353596932078771872, −3.20701839297878657801985375793, −1.64431048596525774752214117343, 3.10401902288404056638358140628, 4.13740398714130101034211350663, 5.79769984959323748999069779376, 6.97289086010954954195260609876, 7.990194051881563543684394312619, 8.886618666231609010602033719552, 9.971072682259200632159903141825, 10.95483851734684427061327599431, 12.89233230837405007147200194659, 13.31559705289449053403408137358

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