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Properties

Label 2-588-21.5-c3-0-4
Degree $2$
Conductor $588$
Sign $-0.951 + 0.307i$
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  + (−5.15 + 0.637i)3-s + (−6.99 + 12.1i)5-s + (26.1 − 6.57i)9-s + (−27.5 + 15.8i)11-s + 31.1i·13-s + (28.3 − 66.9i)15-s + (61.5 + 106. i)17-s + (62.7 + 36.2i)19-s + (66.1 + 38.1i)23-s + (−35.3 − 61.1i)25-s + (−130. + 50.6i)27-s + 14.1i·29-s + (−267. + 154. i)31-s + (131. − 99.5i)33-s + (58.3 − 101. i)37-s + ⋯
L(s)  = 1  + (−0.992 + 0.122i)3-s + (−0.625 + 1.08i)5-s + (0.969 − 0.243i)9-s + (−0.754 + 0.435i)11-s + 0.663i·13-s + (0.487 − 1.15i)15-s + (0.877 + 1.52i)17-s + (0.757 + 0.437i)19-s + (0.599 + 0.346i)23-s + (−0.282 − 0.489i)25-s + (−0.932 + 0.360i)27-s + 0.0903i·29-s + (−1.55 + 0.896i)31-s + (0.695 − 0.525i)33-s + (0.259 − 0.449i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.951 + 0.307i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.951 + 0.307i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6466692835\)
\(L(\frac12)\) \(\approx\) \(0.6466692835\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.15 - 0.637i)T \)
7 \( 1 \)
good5 \( 1 + (6.99 - 12.1i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (27.5 - 15.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 31.1iT - 2.19e3T^{2} \)
17 \( 1 + (-61.5 - 106. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-62.7 - 36.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-66.1 - 38.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 14.1iT - 2.43e4T^{2} \)
31 \( 1 + (267. - 154. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-58.3 + 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 491.T + 6.89e4T^{2} \)
43 \( 1 - 13.6T + 7.95e4T^{2} \)
47 \( 1 + (-43.4 + 75.2i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (388. - 224. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (128. + 222. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (238. + 137. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (515. + 893. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 931. iT - 3.57e5T^{2} \)
73 \( 1 + (887. - 512. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-462. + 800. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 991.T + 5.71e5T^{2} \)
89 \( 1 + (125. - 217. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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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

−10.81665425224111847867023554315, −10.23124681780063828594344602235, −9.216992705721487666347008479742, −7.68349554568192096821269212497, −7.29625885793632760147007121433, −6.21461384375684683661971812518, −5.38324570250523638976638966294, −4.15986089941627328674515633203, −3.22955875634638749289536394003, −1.55083259676870849167723074268, 0.28025841276263810926784491969, 0.999447790778308681964246629946, 2.95680436614765462585231598979, 4.42732508568588780220371374161, 5.20468275566755631085853208512, 5.83018177384273679164857506732, 7.37702601376956934068157706058, 7.75196359285484093819781758886, 9.003354482649497393892840158857, 9.819658233396751160926216027435

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