↑

Properties

Label 2-152-152.125-c1-0-1
Degree $2$
Conductor $152$
Sign $0.728 + 0.684i$
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  + (−1.03 − 0.962i)2-s + (−1.63 − 0.941i)3-s + (0.146 + 1.99i)4-s + (3.38 + 1.95i)5-s + (0.783 + 2.54i)6-s + 0.489·7-s + (1.76 − 2.20i)8-s + (0.273 + 0.473i)9-s + (−1.62 − 5.28i)10-s + 1.19i·11-s + (1.63 − 3.39i)12-s + (3.60 − 2.07i)13-s + (−0.506 − 0.470i)14-s + (−3.68 − 6.37i)15-s + (−3.95 + 0.586i)16-s + (1.07 − 1.87i)17-s + ⋯
L(s)  = 1  + (−0.732 − 0.680i)2-s + (−0.941 − 0.543i)3-s + (0.0734 + 0.997i)4-s + (1.51 + 0.874i)5-s + (0.319 + 1.03i)6-s + 0.184·7-s + (0.624 − 0.780i)8-s + (0.0910 + 0.157i)9-s + (−0.514 − 1.67i)10-s + 0.360i·11-s + (0.472 − 0.978i)12-s + (0.998 − 0.576i)13-s + (−0.135 − 0.125i)14-s + (−0.950 − 1.64i)15-s + (−0.989 + 0.146i)16-s + (0.261 − 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.728 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723561 - 0.286488i\)
\(L(\frac12)\) \(\approx\) \(0.723561 - 0.286488i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 + 0.962i)T \)
19 \( 1 + (-4.24 + 1.00i)T \)
good3 \( 1 + (1.63 + 0.941i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.38 - 1.95i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.489T + 7T^{2} \)
11 \( 1 - 1.19iT - 11T^{2} \)
13 \( 1 + (-3.60 + 2.07i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.07 + 1.87i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.606 + 1.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.31 - 4.80i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 + 8.92iT - 37T^{2} \)
41 \( 1 + (3.32 - 5.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 1.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.95 + 5.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.375 + 0.216i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.71 + 5.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.58 - 0.912i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.58 - 3.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.86 - 11.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.34 - 5.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.39 + 7.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.47iT - 83T^{2} \)
89 \( 1 + (3.15 + 5.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.02 - 1.77i)T + (-48.5 - 84.0i)T^{2} \)
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

−12.78620153194940081545260882858, −11.57448637948122896212067591671, −10.89156637067874614108419847936, −10.00178748071075226182885088796, −9.087528782795169136537978577627, −7.46225631134210482819473065564, −6.48909931000166049353881250093, −5.46902123687285597666886444291, −3.08654531842200111238671070377, −1.49259939897111793392560196327, 1.50178462820889466008832419801, 4.75363145228616403724263540045, 5.74766539839111725384295818367, 6.22044284103299976574663051383, 8.079384336583988655990958253129, 9.172991653406815010758958034512, 9.908583158221272729687688514267, 10.80837396197751075099666926837, 11.82269346798201815215467193619, 13.50562672362311802232891219162

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