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Properties

Label 2-153-17.8-c3-0-6
Degree $2$
Conductor $153$
Sign $0.897 - 0.440i$
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  + (−3.10 + 3.10i)2-s − 11.2i·4-s + (−14.1 − 5.87i)5-s + (−30.3 + 12.5i)7-s + (10.0 + 10.0i)8-s + (62.2 − 25.7i)10-s + (9.27 + 22.4i)11-s + 35.4i·13-s + (55.1 − 133. i)14-s + 27.6·16-s + (27.6 − 64.4i)17-s + (85.2 − 85.2i)19-s + (−66.0 + 159. i)20-s + (−98.2 − 40.7i)22-s + (6.73 + 16.2i)23-s + ⋯
L(s)  = 1  + (−1.09 + 1.09i)2-s − 1.40i·4-s + (−1.26 − 0.525i)5-s + (−1.63 + 0.679i)7-s + (0.443 + 0.443i)8-s + (1.96 − 0.815i)10-s + (0.254 + 0.614i)11-s + 0.757i·13-s + (1.05 − 2.54i)14-s + 0.431·16-s + (0.394 − 0.918i)17-s + (1.02 − 1.02i)19-s + (−0.738 + 1.78i)20-s + (−0.952 − 0.394i)22-s + (0.0610 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.897 - 0.440i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ 0.897 - 0.440i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.387488 + 0.0899145i\)
\(L(\frac12)\) \(\approx\) \(0.387488 + 0.0899145i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-27.6 + 64.4i)T \)
good2 \( 1 + (3.10 - 3.10i)T - 8iT^{2} \)
5 \( 1 + (14.1 + 5.87i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (30.3 - 12.5i)T + (242. - 242. i)T^{2} \)
11 \( 1 + (-9.27 - 22.4i)T + (-941. + 941. i)T^{2} \)
13 \( 1 - 35.4iT - 2.19e3T^{2} \)
19 \( 1 + (-85.2 + 85.2i)T - 6.85e3iT^{2} \)
23 \( 1 + (-6.73 - 16.2i)T + (-8.60e3 + 8.60e3i)T^{2} \)
29 \( 1 + (-148. - 61.5i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + (3.83 - 9.26i)T + (-2.10e4 - 2.10e4i)T^{2} \)
37 \( 1 + (58.4 - 141. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (118. - 48.9i)T + (4.87e4 - 4.87e4i)T^{2} \)
43 \( 1 + (388. + 388. i)T + 7.95e4iT^{2} \)
47 \( 1 + 313. iT - 1.03e5T^{2} \)
53 \( 1 + (47.5 - 47.5i)T - 1.48e5iT^{2} \)
59 \( 1 + (-190. - 190. i)T + 2.05e5iT^{2} \)
61 \( 1 + (343. - 142. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 - 546.T + 3.00e5T^{2} \)
71 \( 1 + (-262. + 633. i)T + (-2.53e5 - 2.53e5i)T^{2} \)
73 \( 1 + (-529. - 219. i)T + (2.75e5 + 2.75e5i)T^{2} \)
79 \( 1 + (-34.7 - 83.7i)T + (-3.48e5 + 3.48e5i)T^{2} \)
83 \( 1 + (106. - 106. i)T - 5.71e5iT^{2} \)
89 \( 1 - 611. iT - 7.04e5T^{2} \)
97 \( 1 + (-437. - 181. i)T + (6.45e5 + 6.45e5i)T^{2} \)
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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

−12.31129275627770021810425093660, −11.79078416377028088404599758611, −9.952449061868131101107358783059, −9.258039582634884678438601090383, −8.499724735182213691140944720816, −7.19596954634703329508041462430, −6.68845424419964799648193599124, −5.11041231972929151450388454568, −3.34820183501052906958751064972, −0.45121948977269598934033713412, 0.76001107283623002841646151527, 3.22138684439417400375877770354, 3.59055354429969834670350262338, 6.21542847284939866727279678700, 7.56541709369024261180424983414, 8.353766189283486587705636297375, 9.748143197984899580941416465633, 10.32355886023300891517773847201, 11.23825176985806390209402851488, 12.17875596993750467727010866872

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