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Properties

Label 2-156-156.107-c1-0-2
Degree $2$
Conductor $156$
Sign $0.999 + 0.0221i$
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  + (−1.06 − 0.926i)2-s + (−1.63 − 0.577i)3-s + (0.283 + 1.97i)4-s + 1.45i·5-s + (1.20 + 2.12i)6-s + (1.17 + 0.680i)7-s + (1.53 − 2.37i)8-s + (2.33 + 1.88i)9-s + (1.34 − 1.55i)10-s + (0.711 + 1.23i)11-s + (0.680 − 3.39i)12-s + (3.27 − 1.50i)13-s + (−0.629 − 1.82i)14-s + (0.838 − 2.36i)15-s + (−3.83 + 1.12i)16-s + (1.68 + 0.971i)17-s + ⋯
L(s)  = 1  + (−0.755 − 0.655i)2-s + (−0.942 − 0.333i)3-s + (0.141 + 0.989i)4-s + 0.648i·5-s + (0.493 + 0.869i)6-s + (0.445 + 0.257i)7-s + (0.541 − 0.840i)8-s + (0.777 + 0.628i)9-s + (0.425 − 0.490i)10-s + (0.214 + 0.371i)11-s + (0.196 − 0.980i)12-s + (0.908 − 0.417i)13-s + (−0.168 − 0.486i)14-s + (0.216 − 0.611i)15-s + (−0.959 + 0.280i)16-s + (0.408 + 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.999 + 0.0221i$
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: \((2,\ 156,\ (\ :1/2),\ 0.999 + 0.0221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644459 - 0.00714660i\)
\(L(\frac12)\) \(\approx\) \(0.644459 - 0.00714660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 + 0.926i)T \)
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

−12.57828201539136642544720876806, −11.72959164944810734407169567246, −10.90607621392269078551167083095, −10.27550593063957599486623734590, −8.926564977268227531590070826185, −7.65524038883460299906520473033, −6.78521019282992453958755906867, −5.33208463429104111641709252502, −3.51881814892569265976116366957, −1.58077944983097293079163332172, 1.07225776657762461827894365776, 4.36060736212182028582804117947, 5.48142744793072884250191921722, 6.46408748571008724921257192923, 7.71653108474799981557891279346, 8.912067645592185330348499583523, 9.753452001996000653957455925512, 11.00620557747933500757793784801, 11.49154879518678243914304094196, 12.92977565026755078818831959112

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