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Properties

Label 2-153-17.15-c3-0-10
Degree $2$
Conductor $153$
Sign $0.816 - 0.577i$
Analytic cond. $9.02729$
Root an. cond. $3.00454$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.89 + 1.89i)2-s − 0.785i·4-s + (−1.92 + 0.798i)5-s + (23.0 + 9.56i)7-s + (16.6 − 16.6i)8-s + (−5.17 − 2.14i)10-s + (−1.46 + 3.53i)11-s − 17.6i·13-s + (25.6 + 62.0i)14-s + 57.1·16-s + (69.7 + 6.56i)17-s + (113. + 113. i)19-s + (0.626 + 1.51i)20-s + (−9.48 + 3.92i)22-s + (38.2 − 92.3i)23-s + ⋯
L(s)  = 1  + (0.671 + 0.671i)2-s − 0.0981i·4-s + (−0.172 + 0.0714i)5-s + (1.24 + 0.516i)7-s + (0.737 − 0.737i)8-s + (−0.163 − 0.0678i)10-s + (−0.0400 + 0.0967i)11-s − 0.377i·13-s + (0.490 + 1.18i)14-s + 0.892·16-s + (0.995 + 0.0936i)17-s + (1.37 + 1.37i)19-s + (0.00700 + 0.0169i)20-s + (−0.0919 + 0.0380i)22-s + (0.346 − 0.837i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.58714 + 0.822533i\)
\(L(\frac12)\) \(\approx\) \(2.58714 + 0.822533i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (-69.7 - 6.56i)T \)
good2 \( 1 + (-1.89 - 1.89i)T + 8iT^{2} \)
5 \( 1 + (1.92 - 0.798i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-23.0 - 9.56i)T + (242. + 242. i)T^{2} \)
11 \( 1 + (1.46 - 3.53i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + 17.6iT - 2.19e3T^{2} \)
19 \( 1 + (-113. - 113. i)T + 6.85e3iT^{2} \)
23 \( 1 + (-38.2 + 92.3i)T + (-8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (185. - 76.7i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (29.2 + 70.7i)T + (-2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (93.6 + 226. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-49.9 - 20.7i)T + (4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (100. - 100. i)T - 7.95e4iT^{2} \)
47 \( 1 + 468. iT - 1.03e5T^{2} \)
53 \( 1 + (68.2 + 68.2i)T + 1.48e5iT^{2} \)
59 \( 1 + (257. - 257. i)T - 2.05e5iT^{2} \)
61 \( 1 + (653. + 270. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 304.T + 3.00e5T^{2} \)
71 \( 1 + (-179. - 434. i)T + (-2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (131. - 54.3i)T + (2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (274. - 663. i)T + (-3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (-259. - 259. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (834. - 345. i)T + (6.45e5 - 6.45e5i)T^{2} \)
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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

−12.70657462882051839569971181441, −11.74520739139165141275239840201, −10.66459890382125419706926346033, −9.565502259966665702116158439778, −8.054895185910907726085642313915, −7.34537484748540823006346556237, −5.71409340206253372141551742318, −5.20179438739502196547344558311, −3.72051608316568961189244467244, −1.52384047431185988825626213261, 1.48258584488545648760520471519, 3.16219143316689699283564401750, 4.43828956906454495249521232899, 5.33875105782634709272228653613, 7.36992900378820282953360339179, 8.010942200868768159323732239992, 9.454801824255095302207278175970, 10.87906245277873706199192286308, 11.50877656839601573567015650423, 12.21207741515735097696130364859

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