Properties

Label 2-588-12.11-c1-0-24
Degree $2$
Conductor $588$
Sign $0.923 + 0.384i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.430i)2-s + (−0.916 − 1.46i)3-s + (1.62 − 1.15i)4-s − 0.348i·5-s + (1.86 + 1.58i)6-s + (−1.69 + 2.26i)8-s + (−1.31 + 2.69i)9-s + (0.150 + 0.469i)10-s + 3.90·11-s + (−3.19 − 1.33i)12-s + 2.93·13-s + (−0.512 + 0.319i)15-s + (1.30 − 3.77i)16-s + 3.90i·17-s + (0.617 − 4.19i)18-s + 5.57i·19-s + ⋯
L(s)  = 1  + (−0.952 + 0.304i)2-s + (−0.529 − 0.848i)3-s + (0.814 − 0.579i)4-s − 0.155i·5-s + (0.762 + 0.647i)6-s + (−0.599 + 0.800i)8-s + (−0.439 + 0.898i)9-s + (0.0474 + 0.148i)10-s + 1.17·11-s + (−0.923 − 0.384i)12-s + 0.815·13-s + (−0.132 + 0.0825i)15-s + (0.327 − 0.944i)16-s + 0.946i·17-s + (0.145 − 0.989i)18-s + 1.27i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.923 + 0.384i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841477 - 0.168178i\)
\(L(\frac12)\) \(\approx\) \(0.841477 - 0.168178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.430i)T \)
3 \( 1 + (0.916 + 1.46i)T \)
7 \( 1 \)
good5 \( 1 + 0.348iT - 5T^{2} \)
11 \( 1 - 3.90T + 11T^{2} \)
13 \( 1 - 2.93T + 13T^{2} \)
17 \( 1 - 3.90iT - 17T^{2} \)
19 \( 1 - 5.57iT - 19T^{2} \)
23 \( 1 + 2.18T + 23T^{2} \)
29 \( 1 + 9.75iT - 29T^{2} \)
31 \( 1 + 2.63iT - 31T^{2} \)
37 \( 1 - 0.639T + 37T^{2} \)
41 \( 1 + 7.57iT - 41T^{2} \)
43 \( 1 - 2.51iT - 43T^{2} \)
47 \( 1 - 4.36T + 47T^{2} \)
53 \( 1 - 1.72iT - 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 - 0.639iT - 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 7.87T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.61987856022080171337965278249, −9.763405943289541827430707475700, −8.600331494346146612292124015765, −8.118552705705232208991558702550, −7.05814228349927325586978121785, −6.19864574782005998444724138180, −5.70733703531021994093981692353, −3.96146835490492033259436920909, −2.08605698565400616673846815585, −1.01214655487073257958045288397, 1.01323756932588548871768424696, 2.93167491781606170369917519936, 3.91978428942513463134435027319, 5.18525656580200964063387141675, 6.54129989633316213576650785525, 6.98927473366095527089903045003, 8.593206489220664910359634975313, 9.060972124532225376115470915802, 9.843202196031929901510600073453, 10.80185870370318531780962295440

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