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Properties

Label 2-153-17.2-c3-0-16
Degree $2$
Conductor $153$
Sign $0.00939 + 0.999i$
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  + (1.20 + 1.20i)2-s − 5.09i·4-s + (−2.60 − 6.29i)5-s + (−5.31 + 12.8i)7-s + (15.7 − 15.7i)8-s + (4.44 − 10.7i)10-s + (−28.4 − 11.8i)11-s − 66.0i·13-s + (−21.8 + 9.05i)14-s − 2.65·16-s + (−3.91 − 69.9i)17-s + (−56.3 − 56.3i)19-s + (−32.0 + 13.2i)20-s + (−20.1 − 48.5i)22-s + (26.6 + 11.0i)23-s + ⋯
L(s)  = 1  + (0.426 + 0.426i)2-s − 0.636i·4-s + (−0.233 − 0.562i)5-s + (−0.286 + 0.692i)7-s + (0.697 − 0.697i)8-s + (0.140 − 0.339i)10-s + (−0.780 − 0.323i)11-s − 1.40i·13-s + (−0.417 + 0.172i)14-s − 0.0414·16-s + (−0.0558 − 0.998i)17-s + (−0.680 − 0.680i)19-s + (−0.358 + 0.148i)20-s + (−0.195 − 0.470i)22-s + (0.242 + 0.100i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00939 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00939 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.00939 + 0.999i$
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: \((2,\ 153,\ (\ :3/2),\ 0.00939 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08580 - 1.07564i\)
\(L(\frac12)\) \(\approx\) \(1.08580 - 1.07564i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.91 + 69.9i)T \)
good2 \( 1 + (-1.20 - 1.20i)T + 8iT^{2} \)
5 \( 1 + (2.60 + 6.29i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (5.31 - 12.8i)T + (-242. - 242. i)T^{2} \)
11 \( 1 + (28.4 + 11.8i)T + (941. + 941. i)T^{2} \)
13 \( 1 + 66.0iT - 2.19e3T^{2} \)
19 \( 1 + (56.3 + 56.3i)T + 6.85e3iT^{2} \)
23 \( 1 + (-26.6 - 11.0i)T + (8.60e3 + 8.60e3i)T^{2} \)
29 \( 1 + (-101. - 245. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + (33.0 - 13.6i)T + (2.10e4 - 2.10e4i)T^{2} \)
37 \( 1 + (-330. + 136. i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (12.5 - 30.2i)T + (-4.87e4 - 4.87e4i)T^{2} \)
43 \( 1 + (364. - 364. i)T - 7.95e4iT^{2} \)
47 \( 1 + 210. iT - 1.03e5T^{2} \)
53 \( 1 + (61.6 + 61.6i)T + 1.48e5iT^{2} \)
59 \( 1 + (219. - 219. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-139. + 337. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 - 660.T + 3.00e5T^{2} \)
71 \( 1 + (-367. + 152. i)T + (2.53e5 - 2.53e5i)T^{2} \)
73 \( 1 + (-246. - 594. i)T + (-2.75e5 + 2.75e5i)T^{2} \)
79 \( 1 + (355. + 147. i)T + (3.48e5 + 3.48e5i)T^{2} \)
83 \( 1 + (108. + 108. i)T + 5.71e5iT^{2} \)
89 \( 1 - 599. iT - 7.04e5T^{2} \)
97 \( 1 + (17.0 + 41.1i)T + (-6.45e5 + 6.45e5i)T^{2} \)
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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

−12.69172136245727300719137458801, −11.19706678880279247250948019803, −10.25926152210607200072842358103, −9.122021290649502832344744814239, −8.034267145417602310852181501772, −6.69311563297052656500091386037, −5.48050949881910264918134953960, −4.81234742656066470481589027836, −2.85417761610707792527095892233, −0.63217503814916052706373984990, 2.19611943799874642721036130634, 3.65296714689534970014997696915, 4.54143707271566186781558261297, 6.44141019187309158729408022575, 7.47220133301924017977454449715, 8.457136396953635365795828392286, 10.00199333881536584167638460712, 10.90831838874235988184830231845, 11.77491098162558087585642890411, 12.80940303696967243769165111336

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