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Properties

Label 2-153-17.15-c3-0-11
Degree $2$
Conductor $153$
Sign $-0.937 - 0.348i$
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  + (−3.81 − 3.81i)2-s + 21.1i·4-s + (−1.33 + 0.554i)5-s + (8.84 + 3.66i)7-s + (50.0 − 50.0i)8-s + (7.22 + 2.99i)10-s + (−17.3 + 41.8i)11-s − 86.3i·13-s + (−19.7 − 47.7i)14-s − 212.·16-s + (−33.9 − 61.3i)17-s + (−25.5 − 25.5i)19-s + (−11.7 − 28.2i)20-s + (225. − 93.4i)22-s + (12.9 − 31.2i)23-s + ⋯
L(s)  = 1  + (−1.34 − 1.34i)2-s + 2.63i·4-s + (−0.119 + 0.0495i)5-s + (0.477 + 0.197i)7-s + (2.21 − 2.21i)8-s + (0.228 + 0.0945i)10-s + (−0.474 + 1.14i)11-s − 1.84i·13-s + (−0.377 − 0.910i)14-s − 3.32·16-s + (−0.484 − 0.875i)17-s + (−0.308 − 0.308i)19-s + (−0.130 − 0.315i)20-s + (2.18 − 0.905i)22-s + (0.117 − 0.283i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.937 - 0.348i$
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: \((2,\ 153,\ (\ :3/2),\ -0.937 - 0.348i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0594237 + 0.330219i\)
\(L(\frac12)\) \(\approx\) \(0.0594237 + 0.330219i\)
\(L(\frac{5}{2})\) 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 \( 1 + (33.9 + 61.3i)T \)
good2 \( 1 + (3.81 + 3.81i)T + 8iT^{2} \)
5 \( 1 + (1.33 - 0.554i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-8.84 - 3.66i)T + (242. + 242. i)T^{2} \)
11 \( 1 + (17.3 - 41.8i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + 86.3iT - 2.19e3T^{2} \)
19 \( 1 + (25.5 + 25.5i)T + 6.85e3iT^{2} \)
23 \( 1 + (-12.9 + 31.2i)T + (-8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (-50.3 + 20.8i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (-52.5 - 126. i)T + (-2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (166. + 401. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (128. + 53.3i)T + (4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (231. - 231. i)T - 7.95e4iT^{2} \)
47 \( 1 + 357. iT - 1.03e5T^{2} \)
53 \( 1 + (118. + 118. i)T + 1.48e5iT^{2} \)
59 \( 1 + (216. - 216. i)T - 2.05e5iT^{2} \)
61 \( 1 + (198. + 82.1i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + 287.T + 3.00e5T^{2} \)
71 \( 1 + (312. + 754. i)T + (-2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (-301. + 124. i)T + (2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-230. + 557. i)T + (-3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (601. + 601. i)T + 5.71e5iT^{2} \)
89 \( 1 - 640. iT - 7.04e5T^{2} \)
97 \( 1 + (-622. + 257. i)T + (6.45e5 - 6.45e5i)T^{2} \)
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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

−11.74951109480005019722606259941, −10.71784335663947240372940868079, −10.10792036620660225545550465886, −9.029251613265213229227531301229, −8.024548329693957487176851250318, −7.24913798655740170135652059518, −4.90363133382532546523599388226, −3.20944298585490337317713464634, −2.00406833193304643026540936299, −0.25123913376458842603042520653, 1.59147520958125974176204236982, 4.54421166815279869654120632000, 5.99606026670883825490931636710, 6.78842688498728404020035709104, 8.093355855093347797220197283892, 8.594659803117958758813543440589, 9.710780639982677766075864883827, 10.75111218658183314723799818171, 11.61700451553911371007549712528, 13.63778067621914450210734873332

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