↑

Properties

Label 2-152-152.75-c1-0-13
Degree $2$
Conductor $152$
Sign $-0.133 + 0.991i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.491 − 1.32i)2-s + 0.754i·3-s + (−1.51 − 1.30i)4-s − 2.08i·5-s + (1 + 0.370i)6-s − 2.23i·7-s + (−2.47 + 1.37i)8-s + 2.43·9-s + (−2.76 − 1.02i)10-s − 0.602·11-s + (0.982 − 1.14i)12-s + 1.29·13-s + (−2.96 − 1.09i)14-s + 1.57·15-s + (0.602 + 3.95i)16-s + 2.20·17-s + ⋯
L(s)  = 1  + (0.347 − 0.937i)2-s + 0.435i·3-s + (−0.758 − 0.651i)4-s − 0.934i·5-s + (0.408 + 0.151i)6-s − 0.845i·7-s + (−0.874 + 0.484i)8-s + 0.810·9-s + (−0.875 − 0.324i)10-s − 0.181·11-s + (0.283 − 0.330i)12-s + 0.359·13-s + (−0.792 − 0.293i)14-s + 0.406·15-s + (0.150 + 0.988i)16-s + 0.534·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.133 + 0.991i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.133 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819397 - 0.937028i\)
\(L(\frac12)\) \(\approx\) \(0.819397 - 0.937028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.491 + 1.32i)T \)
19 \( 1 + (2.60 - 3.49i)T \)
good3 \( 1 - 0.754iT - 3T^{2} \)
5 \( 1 + 2.08iT - 5T^{2} \)
7 \( 1 + 2.23iT - 7T^{2} \)
11 \( 1 + 0.602T + 11T^{2} \)
13 \( 1 - 1.29T + 13T^{2} \)
17 \( 1 - 2.20T + 17T^{2} \)
23 \( 1 - 6.19iT - 23T^{2} \)
29 \( 1 + 8.20T + 29T^{2} \)
31 \( 1 - 4.94T + 31T^{2} \)
37 \( 1 - 6.91T + 37T^{2} \)
41 \( 1 + 6.53iT - 41T^{2} \)
43 \( 1 + 0.191T + 43T^{2} \)
47 \( 1 - 0.223iT - 47T^{2} \)
53 \( 1 + 4.83T + 53T^{2} \)
59 \( 1 + 5.60iT - 59T^{2} \)
61 \( 1 - 11.7iT - 61T^{2} \)
67 \( 1 + 6.23iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 8.27T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 7.44iT - 89T^{2} \)
97 \( 1 - 17.6iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.86467238139947487800576822675, −11.74987986945764513651194017499, −10.62619775345373095870098705051, −9.892525981430369545325916941861, −8.952337539001952514743487742608, −7.60493299219672069042240902268, −5.73733287039432734139483405612, −4.53068082412825169969255610087, −3.68925899749569067221896621208, −1.37884138731319244440393907220, 2.79293148298529568667643952077, 4.46323300962751062202782564014, 5.97824227559216108471179333755, 6.78473533674838398708409798546, 7.76863075865256490109849373093, 8.882596223234786853778538277567, 10.11336723477964934869521665060, 11.44660812589026750076984440729, 12.63938654234220350820687296576, 13.22469991580347004574451578416

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