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Properties

Label 2-588-7.4-c5-0-22
Degree $2$
Conductor $588$
Sign $0.605 + 0.795i$
Analytic cond. $94.3056$
Root an. cond. $9.71111$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)3-s + (39.3 + 68.1i)5-s + (−40.5 − 70.1i)9-s + (−345. + 598. i)11-s − 818.·13-s − 708.·15-s + (554. − 960. i)17-s + (−286. − 496. i)19-s + (−1.25e3 − 2.18e3i)23-s + (−1.53e3 + 2.65e3i)25-s + 729·27-s − 3.25e3·29-s + (5.05e3 − 8.76e3i)31-s + (−3.11e3 − 5.38e3i)33-s + (−2.43e3 − 4.21e3i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.703 + 1.21i)5-s + (−0.166 − 0.288i)9-s + (−0.861 + 1.49i)11-s − 1.34·13-s − 0.812·15-s + (0.465 − 0.806i)17-s + (−0.182 − 0.315i)19-s + (−0.496 − 0.859i)23-s + (−0.490 + 0.849i)25-s + 0.192·27-s − 0.719·29-s + (0.945 − 1.63i)31-s + (−0.497 − 0.861i)33-s + (−0.292 − 0.506i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6519688496\)
\(L(\frac12)\) \(\approx\) \(0.6519688496\)
\(L(\frac{7}{2})\) 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 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (-39.3 - 68.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (345. - 598. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 818.T + 3.71e5T^{2} \)
17 \( 1 + (-554. + 960. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (286. + 496. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.25e3 + 2.18e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 3.25e3T + 2.05e7T^{2} \)
31 \( 1 + (-5.05e3 + 8.76e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.43e3 + 4.21e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.30e4T + 1.15e8T^{2} \)
43 \( 1 + 9.30e3T + 1.47e8T^{2} \)
47 \( 1 + (-6.45e3 - 1.11e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-9.77e3 + 1.69e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.25e4 - 2.17e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.56e4 + 2.71e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.79e4 - 4.84e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 2.05e4T + 1.80e9T^{2} \)
73 \( 1 + (3.38e4 - 5.85e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (7.03e3 + 1.21e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 7.71e4T + 3.93e9T^{2} \)
89 \( 1 + (-160. - 277. i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.12e5T + 8.58e9T^{2} \)
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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

−9.998486741416113647930281721139, −9.347570929574025191321589420179, −7.69390171999171770428906054037, −7.15591689020391763104503948984, −6.14733292967741216363395231288, −5.13955098848657169366369657713, −4.29459992638884569441479682175, −2.68181713973450480556465538386, −2.27216269510107732894365325685, −0.16350537271178732751894324320, 0.937635321912667336934340148633, 1.93353275365786013877437808135, 3.22921224745443967011301324273, 4.79614377797507738807053062380, 5.51434185144685156469246146685, 6.15341726245543779962908989374, 7.54642164592174511153403678563, 8.277784635628010683501585053747, 9.069166137143647470921573014543, 10.07063365993961363091190051477

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