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Properties

Label 2-153-17.2-c3-0-6
Degree $2$
Conductor $153$
Sign $-0.139 - 0.990i$
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 − 0.828i·4-s + (1.56 + 3.77i)5-s + (−11.5 + 27.9i)7-s + (16.7 − 16.7i)8-s + (−4.19 + 10.1i)10-s + (31.0 + 12.8i)11-s + 69.9i·13-s + (−74.8 + 30.9i)14-s + 56.6·16-s + (13.0 + 68.8i)17-s + (−60.7 − 60.7i)19-s + (3.13 − 1.29i)20-s + (34.4 + 83.2i)22-s + (117. + 48.6i)23-s + ⋯
L(s)  = 1  + (0.669 + 0.669i)2-s − 0.103i·4-s + (0.140 + 0.338i)5-s + (−0.624 + 1.50i)7-s + (0.738 − 0.738i)8-s + (−0.132 + 0.320i)10-s + (0.851 + 0.352i)11-s + 1.49i·13-s + (−1.42 + 0.591i)14-s + 0.885·16-s + (0.185 + 0.982i)17-s + (−0.733 − 0.733i)19-s + (0.0349 − 0.0144i)20-s + (0.334 + 0.806i)22-s + (1.06 + 0.440i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ -0.139 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.51505 + 1.74368i\)
\(L(\frac12)\) \(\approx\) \(1.51505 + 1.74368i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.0 - 68.8i)T \)
good2 \( 1 + (-1.89 - 1.89i)T + 8iT^{2} \)
5 \( 1 + (-1.56 - 3.77i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (11.5 - 27.9i)T + (-242. - 242. i)T^{2} \)
11 \( 1 + (-31.0 - 12.8i)T + (941. + 941. i)T^{2} \)
13 \( 1 - 69.9iT - 2.19e3T^{2} \)
19 \( 1 + (60.7 + 60.7i)T + 6.85e3iT^{2} \)
23 \( 1 + (-117. - 48.6i)T + (8.60e3 + 8.60e3i)T^{2} \)
29 \( 1 + (46.0 + 111. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + (208. - 86.2i)T + (2.10e4 - 2.10e4i)T^{2} \)
37 \( 1 + (60.7 - 25.1i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-4.51 + 10.9i)T + (-4.87e4 - 4.87e4i)T^{2} \)
43 \( 1 + (-180. + 180. i)T - 7.95e4iT^{2} \)
47 \( 1 + 478. iT - 1.03e5T^{2} \)
53 \( 1 + (-187. - 187. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-24.4 + 24.4i)T - 2.05e5iT^{2} \)
61 \( 1 + (13.9 - 33.5i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 - 594.T + 3.00e5T^{2} \)
71 \( 1 + (-794. + 328. i)T + (2.53e5 - 2.53e5i)T^{2} \)
73 \( 1 + (172. + 416. i)T + (-2.75e5 + 2.75e5i)T^{2} \)
79 \( 1 + (-993. - 411. i)T + (3.48e5 + 3.48e5i)T^{2} \)
83 \( 1 + (68.2 + 68.2i)T + 5.71e5iT^{2} \)
89 \( 1 + 431. iT - 7.04e5T^{2} \)
97 \( 1 + (-192. - 463. i)T + (-6.45e5 + 6.45e5i)T^{2} \)
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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

−12.86713694997582717229425145504, −12.04693343822682652175130801300, −10.83801380777117829509786237917, −9.488073191035635505245823325461, −8.826495077727302047650508335277, −6.89113515010990314421033514733, −6.39002905536065681954342245014, −5.26931445745528192602859520012, −3.91096877795897271883964752520, −2.03841539656893198224864211687, 0.972207518731993740981186619248, 3.10996472240906481040231046522, 3.99363173202378377411972753459, 5.29760295195378446903857269092, 6.91897620451861656977976574222, 7.933929254613539154890092666017, 9.325832938723423223042607202837, 10.58560995300952341146321532015, 11.15163282775111226712776229604, 12.68541218064030715624038974472

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