Properties

Label 2-608-152.11-c0-0-0
Degree $2$
Conductor $608$
Sign $0.305 - 0.952i$
Analytic cond. $0.303431$
Root an. cond. $0.550846$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + 11-s + (−1 + 1.73i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)41-s + (1 − 1.73i)43-s + 49-s + (−0.999 − 1.73i)51-s − 0.999·57-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)67-s + (0.5 − 0.866i)73-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 11-s + (−1 + 1.73i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)41-s + (1 − 1.73i)43-s + 49-s + (−0.999 − 1.73i)51-s − 0.999·57-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)67-s + (0.5 − 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(0.303431\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :0),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8042628884\)
\(L(\frac12)\) \(\approx\) \(0.8042628884\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.72838366275460956136030641639, −10.41801524758241042228149675412, −9.369956775774070160574742727137, −8.592155664383104785485620073556, −7.50191829759378837732288657574, −6.30017884713119662982360887260, −5.61262967968093306371810311304, −4.29070765762225192662841796841, −3.83026987453406008490100688988, −1.92319419153140999332962519896, 1.12585683036847704027328366797, 2.67437094137505263442893173742, 4.14714638658681278038487665856, 5.27092499718470067659696579444, 6.41371965965863800961026966213, 6.98546810395350658811309259632, 7.77289761096655262451228694979, 9.265106896335518555685611048344, 9.441622027031549088590967758695, 11.10447021548712800271749983443

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