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Properties

Label 2-153-17.9-c3-0-11
Degree $2$
Conductor $153$
Sign $0.974 - 0.222i$
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  + (1.56 − 1.56i)2-s + 3.11i·4-s + (3.46 − 8.37i)5-s + (11.4 + 27.5i)7-s + (17.3 + 17.3i)8-s + (−7.66 − 18.5i)10-s + (−23.8 + 9.88i)11-s + 13.2i·13-s + (60.9 + 25.2i)14-s + 29.3·16-s + (68.9 − 12.3i)17-s + (35.0 − 35.0i)19-s + (26.1 + 10.8i)20-s + (−21.8 + 52.7i)22-s + (174. − 72.2i)23-s + ⋯
L(s)  = 1  + (0.552 − 0.552i)2-s + 0.389i·4-s + (0.310 − 0.749i)5-s + (0.617 + 1.48i)7-s + (0.767 + 0.767i)8-s + (−0.242 − 0.585i)10-s + (−0.654 + 0.270i)11-s + 0.283i·13-s + (1.16 + 0.482i)14-s + 0.458·16-s + (0.984 − 0.176i)17-s + (0.423 − 0.423i)19-s + (0.292 + 0.120i)20-s + (−0.211 + 0.510i)22-s + (1.58 − 0.655i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.974 - 0.222i$
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: \((2,\ 153,\ (\ :3/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.48234 + 0.279557i\)
\(L(\frac12)\) \(\approx\) \(2.48234 + 0.279557i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-68.9 + 12.3i)T \)
good2 \( 1 + (-1.56 + 1.56i)T - 8iT^{2} \)
5 \( 1 + (-3.46 + 8.37i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-11.4 - 27.5i)T + (-242. + 242. i)T^{2} \)
11 \( 1 + (23.8 - 9.88i)T + (941. - 941. i)T^{2} \)
13 \( 1 - 13.2iT - 2.19e3T^{2} \)
19 \( 1 + (-35.0 + 35.0i)T - 6.85e3iT^{2} \)
23 \( 1 + (-174. + 72.2i)T + (8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (64.2 - 155. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (292. + 121. i)T + (2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (-170. - 70.6i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (35.7 + 86.3i)T + (-4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (209. + 209. i)T + 7.95e4iT^{2} \)
47 \( 1 + 294. iT - 1.03e5T^{2} \)
53 \( 1 + (134. - 134. i)T - 1.48e5iT^{2} \)
59 \( 1 + (536. + 536. i)T + 2.05e5iT^{2} \)
61 \( 1 + (3.38 + 8.16i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 635.T + 3.00e5T^{2} \)
71 \( 1 + (713. + 295. i)T + (2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (202. - 488. i)T + (-2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (81.6 - 33.8i)T + (3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (-678. + 678. i)T - 5.71e5iT^{2} \)
89 \( 1 + 3.21iT - 7.04e5T^{2} \)
97 \( 1 + (-135. + 326. i)T + (-6.45e5 - 6.45e5i)T^{2} \)
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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

−12.63560815498014864924835149566, −11.75827916469706608708090610088, −10.90404596685041234795690909336, −9.279099051930161070645794342124, −8.569613981093767071210473236508, −7.39983342538040785259093730204, −5.37881639355680814283417422356, −4.94734026971880022683533859639, −3.09516447964721146261833236232, −1.84300152643512177692023978183, 1.17826773833871704881163041690, 3.42355070885561306807295129097, 4.83220245455226029528148391138, 5.89361447778351389213860812415, 7.16407214242787476933309697299, 7.75091049902569351079719395721, 9.702673948386409589655089278954, 10.58174690559129263796370715424, 11.10859604573927865835485516239, 12.94021214896145990375913663993

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