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Properties

Label 2-153-17.15-c3-0-20
Degree $2$
Conductor $153$
Sign $-0.935 - 0.354i$
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  + (0.161 + 0.161i)2-s − 7.94i·4-s + (−2.54 + 1.05i)5-s + (−19.8 − 8.23i)7-s + (2.56 − 2.56i)8-s + (−0.579 − 0.239i)10-s + (−20.8 + 50.3i)11-s + 52.4i·13-s + (−1.87 − 4.53i)14-s − 62.7·16-s + (−33.5 − 61.5i)17-s + (13.8 + 13.8i)19-s + (8.36 + 20.2i)20-s + (−11.4 + 4.75i)22-s + (5.33 − 12.8i)23-s + ⋯
L(s)  = 1  + (0.0569 + 0.0569i)2-s − 0.993i·4-s + (−0.227 + 0.0941i)5-s + (−1.07 − 0.444i)7-s + (0.113 − 0.113i)8-s + (−0.0183 − 0.00758i)10-s + (−0.572 + 1.38i)11-s + 1.11i·13-s + (−0.0358 − 0.0864i)14-s − 0.980·16-s + (−0.478 − 0.878i)17-s + (0.167 + 0.167i)19-s + (0.0935 + 0.225i)20-s + (−0.111 + 0.0460i)22-s + (0.0483 − 0.116i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.935 - 0.354i$
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: \((2,\ 153,\ (\ :3/2),\ -0.935 - 0.354i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00435936 + 0.0237920i\)
\(L(\frac12)\) \(\approx\) \(0.00435936 + 0.0237920i\)
\(L(\frac{5}{2})\) 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 \( 1 + (33.5 + 61.5i)T \)
good2 \( 1 + (-0.161 - 0.161i)T + 8iT^{2} \)
5 \( 1 + (2.54 - 1.05i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (19.8 + 8.23i)T + (242. + 242. i)T^{2} \)
11 \( 1 + (20.8 - 50.3i)T + (-941. - 941. i)T^{2} \)
13 \( 1 - 52.4iT - 2.19e3T^{2} \)
19 \( 1 + (-13.8 - 13.8i)T + 6.85e3iT^{2} \)
23 \( 1 + (-5.33 + 12.8i)T + (-8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (64.6 - 26.7i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (63.9 + 154. i)T + (-2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (76.0 + 183. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (401. + 166. i)T + (4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (-89.9 + 89.9i)T - 7.95e4iT^{2} \)
47 \( 1 - 207. iT - 1.03e5T^{2} \)
53 \( 1 + (-220. - 220. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-407. + 407. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-72.4 - 29.9i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 359.T + 3.00e5T^{2} \)
71 \( 1 + (81.5 + 196. i)T + (-2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (-26.8 + 11.1i)T + (2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-327. + 790. i)T + (-3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (9.67 + 9.67i)T + 5.71e5iT^{2} \)
89 \( 1 - 651. iT - 7.04e5T^{2} \)
97 \( 1 + (1.10e3 - 458. i)T + (6.45e5 - 6.45e5i)T^{2} \)
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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

−11.92173413214917982154219955849, −10.81776483583120012295885156083, −9.781177585751329423909221766063, −9.286761130826104429830563972382, −7.31990852456460659836913202167, −6.68729147335642560351097498915, −5.26901985186835903599629163055, −4.02979697545569547495695498347, −2.08318463217322256718135145331, −0.01059253751621469844543026243, 2.83871545906268546927564944053, 3.68207029793796593682328201666, 5.49505862752697407030992234578, 6.67905512010970944812572851800, 8.112829581628751931169635276177, 8.646796524251777112449714274889, 10.07369867708807079228771494091, 11.18294243600554712740532922365, 12.22524940181318950014439506023, 13.05966669387213169225034016138

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