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Properties

Label 2-588-7.3-c6-0-30
Degree $2$
Conductor $588$
Sign $0.997 + 0.0633i$
Analytic cond. $135.271$
Root an. cond. $11.6306$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 + 7.79i)3-s + (181. − 104. i)5-s + (121.5 + 210. i)9-s + (−460. + 798. i)11-s + 1.79e3i·13-s + 3.26e3·15-s + (−7.21e3 − 4.16e3i)17-s + (1.71e3 − 992. i)19-s + (−5.34e3 − 9.25e3i)23-s + (1.41e4 − 2.45e4i)25-s + 3.78e3i·27-s + 3.97e4·29-s + (1.58e4 + 9.15e3i)31-s + (−1.24e4 + 7.18e3i)33-s + (2.91e4 + 5.04e4i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (1.45 − 0.838i)5-s + (0.166 + 0.288i)9-s + (−0.346 + 0.599i)11-s + 0.816i·13-s + 0.968·15-s + (−1.46 − 0.847i)17-s + (0.250 − 0.144i)19-s + (−0.439 − 0.760i)23-s + (0.905 − 1.56i)25-s + 0.192i·27-s + 1.62·29-s + (0.531 + 0.307i)31-s + (−0.346 + 0.199i)33-s + (0.575 + 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.997 + 0.0633i$
Analytic conductor: \(135.271\)
Root analytic conductor: \(11.6306\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3),\ 0.997 + 0.0633i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.759359731\)
\(L(\frac12)\) \(\approx\) \(3.759359731\)
\(L(4)\) 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 \( 1 + (-13.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (-181. + 104. i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (460. - 798. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 1.79e3iT - 4.82e6T^{2} \)
17 \( 1 + (7.21e3 + 4.16e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.71e3 + 992. i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (5.34e3 + 9.25e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 3.97e4T + 5.94e8T^{2} \)
31 \( 1 + (-1.58e4 - 9.15e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-2.91e4 - 5.04e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 7.33e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.36e5T + 6.32e9T^{2} \)
47 \( 1 + (-2.09e4 + 1.20e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.22e4 - 2.12e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-5.00e4 - 2.89e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-3.29e5 + 1.90e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-9.75e4 + 1.68e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 2.26e5T + 1.28e11T^{2} \)
73 \( 1 + (-2.80e5 - 1.61e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (2.77e5 + 4.81e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 5.29e5iT - 3.26e11T^{2} \)
89 \( 1 + (-4.11e4 + 2.37e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.02e6iT - 8.32e11T^{2} \)
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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

−9.661223485259010882694712094771, −8.985555985255782699972360623316, −8.281108259175601895317589063029, −6.90155184661683031630955819326, −6.15037538318154353924613578574, −4.79805884112890048144541876133, −4.54537870327685604540275643249, −2.65909944821041613587501214139, −2.05895996864196103188772280955, −0.851561241444661377813392034620, 0.873695010011590240311094992471, 2.19494686717799079044165311817, 2.69298063277582119908832675515, 3.92974812038999819857600443473, 5.48432887804690619212588546319, 6.13968040707841741836192858459, 6.95208371303658005651162985458, 8.041490380967858159942202459942, 8.896821092146831201855224591276, 9.811796373022298445057223074756

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