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Properties

Label 2-153-17.9-c3-0-19
Degree $2$
Conductor $153$
Sign $-0.770 + 0.636i$
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  + (3.43 − 3.43i)2-s − 15.6i·4-s + (7.10 − 17.1i)5-s + (5.36 + 12.9i)7-s + (−26.2 − 26.2i)8-s + (−34.5 − 83.3i)10-s + (−17.9 + 7.45i)11-s + 29.9i·13-s + (62.9 + 26.0i)14-s − 55.3·16-s + (56.0 − 42.0i)17-s + (−32.3 + 32.3i)19-s + (−268. − 111. i)20-s + (−36.2 + 87.4i)22-s + (−82.5 + 34.1i)23-s + ⋯
L(s)  = 1  + (1.21 − 1.21i)2-s − 1.95i·4-s + (0.635 − 1.53i)5-s + (0.289 + 0.699i)7-s + (−1.16 − 1.16i)8-s + (−1.09 − 2.63i)10-s + (−0.493 + 0.204i)11-s + 0.638i·13-s + (1.20 + 0.497i)14-s − 0.865·16-s + (0.799 − 0.600i)17-s + (−0.390 + 0.390i)19-s + (−2.99 − 1.24i)20-s + (−0.351 + 0.847i)22-s + (−0.748 + 0.309i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.636i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.770 + 0.636i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ -0.770 + 0.636i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14854 - 3.19372i\)
\(L(\frac12)\) \(\approx\) \(1.14854 - 3.19372i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-56.0 + 42.0i)T \)
good2 \( 1 + (-3.43 + 3.43i)T - 8iT^{2} \)
5 \( 1 + (-7.10 + 17.1i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-5.36 - 12.9i)T + (-242. + 242. i)T^{2} \)
11 \( 1 + (17.9 - 7.45i)T + (941. - 941. i)T^{2} \)
13 \( 1 - 29.9iT - 2.19e3T^{2} \)
19 \( 1 + (32.3 - 32.3i)T - 6.85e3iT^{2} \)
23 \( 1 + (82.5 - 34.1i)T + (8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (-22.1 + 53.4i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (-149. - 61.9i)T + (2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (-125. - 51.8i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-21.7 - 52.5i)T + (-4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (-36.8 - 36.8i)T + 7.95e4iT^{2} \)
47 \( 1 - 482. iT - 1.03e5T^{2} \)
53 \( 1 + (-374. + 374. i)T - 1.48e5iT^{2} \)
59 \( 1 + (198. + 198. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-224. - 541. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + 367.T + 3.00e5T^{2} \)
71 \( 1 + (-55.9 - 23.1i)T + (2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (140. - 340. i)T + (-2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-201. + 83.3i)T + (3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (420. - 420. i)T - 5.71e5iT^{2} \)
89 \( 1 + 887. iT - 7.04e5T^{2} \)
97 \( 1 + (-338. + 817. i)T + (-6.45e5 - 6.45e5i)T^{2} \)
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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

−12.17745793472906677422446102516, −11.65223744706259657320362238788, −10.17816169483018325759842348183, −9.396579575993357458212548104971, −8.192954191639621187336682355917, −5.93016613979922003866062664021, −5.15412914515488238379028698051, −4.27440867240562826458899818012, −2.46415908031619556691069470176, −1.29551173051216310299966057325, 2.79490022314288842611266018592, 4.00023650876041296610517886531, 5.51655969753273430533218379292, 6.35536631580808082245743513054, 7.28629006142288597686087829547, 8.116209186628264530245146178849, 10.14537119001287666829286955964, 10.81855958100183973251856756905, 12.27360176497015209573288861990, 13.46645264014564476271029436452

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