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Properties

Label 2-588-7.2-c5-0-19
Degree $2$
Conductor $588$
Sign $-0.266 + 0.963i$
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 + (−38.8 + 67.2i)5-s + (−40.5 + 70.1i)9-s + (−238. − 413. i)11-s + 63.7·13-s + 698.·15-s + (518. + 898. i)17-s + (−333. + 577. i)19-s + (−1.62e3 + 2.81e3i)23-s + (−1.45e3 − 2.51e3i)25-s + 729·27-s + 2.30e3·29-s + (1.85e3 + 3.21e3i)31-s + (−2.14e3 + 3.72e3i)33-s + (−6.12e3 + 1.06e4i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.694 + 1.20i)5-s + (−0.166 + 0.288i)9-s + (−0.594 − 1.03i)11-s + 0.104·13-s + 0.801·15-s + (0.435 + 0.754i)17-s + (−0.211 + 0.367i)19-s + (−0.640 + 1.10i)23-s + (−0.464 − 0.803i)25-s + 0.192·27-s + 0.508·29-s + (0.347 + 0.601i)31-s + (−0.343 + 0.594i)33-s + (−0.735 + 1.27i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3856594792\)
\(L(\frac12)\) \(\approx\) \(0.3856594792\)
\(L(\frac{7}{2})\) 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 \( 1 + (38.8 - 67.2i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (238. + 413. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 63.7T + 3.71e5T^{2} \)
17 \( 1 + (-518. - 898. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (333. - 577. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.62e3 - 2.81e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.30e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.85e3 - 3.21e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (6.12e3 - 1.06e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.82e3T + 1.15e8T^{2} \)
43 \( 1 + 2.07e4T + 1.47e8T^{2} \)
47 \( 1 + (2.14e3 - 3.70e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.28e4 + 2.22e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.41e3 + 2.45e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-8.40e3 + 1.45e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.12e4 - 5.41e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 7.23e4T + 1.80e9T^{2} \)
73 \( 1 + (2.78e4 + 4.82e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.99e3 + 3.45e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 4.60e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.76e4 + 1.17e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.42e5T + 8.58e9T^{2} \)
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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

−9.969679165019472652208172332690, −8.340107497236236323854275732585, −7.975738278175087615507194311913, −6.88635258917440491606022157397, −6.21697453981815996563972831298, −5.19916182954355879991555913375, −3.64750509943567458252656660484, −3.02498987225598824498461885088, −1.59804112156089165665430964074, −0.11862014060307787771893686478, 0.799937453075725890137391300786, 2.33762802900758455706464986772, 3.82673394934010298392069978200, 4.71910719493133869426373524742, 5.21062515158322478743735948884, 6.56699996784460935574517593423, 7.69211452236991929118294298481, 8.426269252361693004690008995373, 9.330418730424216858356807112519, 10.10040851420790273398255029053

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