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Properties

Label 2-588-12.11-c1-0-35
Degree $2$
Conductor $588$
Sign $-0.623 + 0.781i$
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  + (−0.866 − 1.11i)2-s + (−1.58 − 0.707i)3-s + (−0.500 + 1.93i)4-s − 2.44i·5-s + (0.578 + 2.38i)6-s + (2.59 − 1.11i)8-s + (2.00 + 2.23i)9-s + (−2.73 + 2.12i)10-s + 3.46·11-s + (2.15 − 2.70i)12-s + 5.47·13-s + (−1.73 + 3.87i)15-s + (−3.5 − 1.93i)16-s − 4.89i·17-s + (0.767 − 4.17i)18-s + 4.24i·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.790i)2-s + (−0.912 − 0.408i)3-s + (−0.250 + 0.968i)4-s − 1.09i·5-s + (0.236 + 0.971i)6-s + (0.918 − 0.395i)8-s + (0.666 + 0.745i)9-s + (−0.866 + 0.670i)10-s + 1.04·11-s + (0.623 − 0.781i)12-s + 1.51·13-s + (−0.447 + 0.999i)15-s + (−0.875 − 0.484i)16-s − 1.18i·17-s + (0.181 − 0.983i)18-s + 0.973i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.623 + 0.781i$
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: \((2,\ 588,\ (\ :1/2),\ -0.623 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363854 - 0.755567i\)
\(L(\frac12)\) \(\approx\) \(0.363854 - 0.755567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 1.11i)T \)
3 \( 1 + (1.58 + 0.707i)T \)
7 \( 1 \)
good5 \( 1 + 2.44iT - 5T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 7.74iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 4.47iT - 53T^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 - 9.79iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.53051947261554251405098283571, −9.474330835745778827626918171777, −8.799121936671760350321131230302, −7.920386583232763437780254286306, −6.85832760334936940106056344445, −5.77212866239104754682356852745, −4.63549487106539622557994473164, −3.67357088040230140214511148401, −1.71084682916194469209399284819, −0.801196028497204132373535914887, 1.33258829372960725565371274200, 3.54791071749851257611475941054, 4.63567885465901024786134450878, 6.04797357748471929800677221002, 6.37652027195887687829112197561, 7.14328299932412820133835637464, 8.440820041400954308701049439361, 9.257454245265769166781952060800, 10.21975090750974765068526747148, 10.97186089660600431943942948573

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