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Properties

Label 2-588-7.4-c3-0-14
Degree $2$
Conductor $588$
Sign $-0.198 + 0.980i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
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  + (−1.5 + 2.59i)3-s + (8.32 + 14.4i)5-s + (−4.5 − 7.79i)9-s + (35.8 − 62.1i)11-s − 65.3·13-s − 49.9·15-s + (−45.2 + 78.3i)17-s + (−81.9 − 141. i)19-s + (−39.6 − 68.6i)23-s + (−76.1 + 131. i)25-s + 27·27-s − 43.2·29-s + (−67.8 + 117. i)31-s + (107. + 186. i)33-s + (−135. − 234. i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.744 + 1.29i)5-s + (−0.166 − 0.288i)9-s + (0.983 − 1.70i)11-s − 1.39·13-s − 0.860·15-s + (−0.645 + 1.11i)17-s + (−0.989 − 1.71i)19-s + (−0.359 − 0.622i)23-s + (−0.609 + 1.05i)25-s + 0.192·27-s − 0.277·29-s + (−0.392 + 0.680i)31-s + (0.567 + 0.983i)33-s + (−0.601 − 1.04i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5498208400\)
\(L(\frac12)\) \(\approx\) \(0.5498208400\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (-8.32 - 14.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-35.8 + 62.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 65.3T + 2.19e3T^{2} \)
17 \( 1 + (45.2 - 78.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (81.9 + 141. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (39.6 + 68.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 43.2T + 2.43e4T^{2} \)
31 \( 1 + (67.8 - 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (135. + 234. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 152.T + 6.89e4T^{2} \)
43 \( 1 + 177.T + 7.95e4T^{2} \)
47 \( 1 + (-22.8 - 39.5i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-79.2 + 137. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-195. + 339. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (275. + 477. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (229. - 397. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 486.T + 3.57e5T^{2} \)
73 \( 1 + (287. - 497. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-334. - 578. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 76.2T + 5.71e5T^{2} \)
89 \( 1 + (683. + 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 242.T + 9.12e5T^{2} \)
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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

−10.23220890463662459764095014935, −9.221713173179549515290329820507, −8.532886676556907463759047335704, −6.96961058565166656378795062700, −6.45510139798978454427596916577, −5.61027139137420899295309226523, −4.30787548434623505382908290082, −3.17768780654672556259011552841, −2.17614906799218996814774532494, −0.15561864794983699068800063103, 1.49797911245287653358479195802, 2.16327804652080286130822874748, 4.27885961625385116146457031242, 4.94359663245313825801844395556, 5.93489433686636956014361960092, 6.98853722292375535431795540384, 7.76491712151086217305168503262, 8.986278999710259840337043758713, 9.608101720623422767096783860879, 10.24632176429651358081249204068

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