↑

Properties

Label 2-152-152.107-c1-0-8
Degree $2$
Conductor $152$
Sign $0.832 + 0.554i$
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  + (−1.40 + 0.175i)2-s + (0.179 − 0.103i)3-s + (1.93 − 0.492i)4-s + (0.520 − 0.300i)5-s + (−0.233 + 0.176i)6-s − 4.27i·7-s + (−2.63 + 1.03i)8-s + (−1.47 + 2.56i)9-s + (−0.677 + 0.512i)10-s + 5.11·11-s + (0.296 − 0.288i)12-s + (1.55 − 2.68i)13-s + (0.749 + 5.99i)14-s + (0.0621 − 0.107i)15-s + (3.51 − 1.90i)16-s + (−1.52 − 2.64i)17-s + ⋯
L(s)  = 1  + (−0.992 + 0.123i)2-s + (0.103 − 0.0597i)3-s + (0.969 − 0.246i)4-s + (0.232 − 0.134i)5-s + (−0.0952 + 0.0720i)6-s − 1.61i·7-s + (−0.931 + 0.364i)8-s + (−0.492 + 0.853i)9-s + (−0.214 + 0.162i)10-s + 1.54·11-s + (0.0855 − 0.0833i)12-s + (0.430 − 0.745i)13-s + (0.200 + 1.60i)14-s + (0.0160 − 0.0277i)15-s + (0.878 − 0.476i)16-s + (−0.371 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781167 - 0.236537i\)
\(L(\frac12)\) \(\approx\) \(0.781167 - 0.236537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.175i)T \)
19 \( 1 + (-3.31 + 2.83i)T \)
good3 \( 1 + (-0.179 + 0.103i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.520 + 0.300i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.27iT - 7T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 + (-1.55 + 2.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.52 + 2.64i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.65 - 2.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.77 - 6.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.59T + 31T^{2} \)
37 \( 1 - 4.44T + 37T^{2} \)
41 \( 1 + (0.253 - 0.146i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.892 + 1.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.79 + 3.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.16 - 8.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.849 - 0.490i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.08 - 2.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.98 - 3.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.173 - 0.300i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.61 - 7.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.63 - 6.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.38T + 83T^{2} \)
89 \( 1 + (-5.44 - 3.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.87 + 4.54i)T + (48.5 - 84.0i)T^{2} \)
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

−13.03728740659527197294534394755, −11.23220001943126532286388446022, −11.05072610701613811032313977041, −9.680887913963688715908855042047, −8.898441568883448195509229213753, −7.52063464445330835140711032091, −6.95619401395860538735540988411, −5.36506712795496773730000770739, −3.45395518808363252572163450379, −1.28691815677632222038925836353, 1.95689093405746220743541154126, 3.56143721855567742146371016421, 5.97294448859171250489717784982, 6.53520373794239840742233352567, 8.274139363260754552978356038411, 9.195927047787800481813335903680, 9.500894002985063357182596440299, 11.29395040307642246411733621222, 11.77486486519648611706406923035, 12.67101928865302409787044484168

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