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Properties

Label 2-588-28.27-c1-0-2
Degree $2$
Conductor $588$
Sign $-0.712 + 0.701i$
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  + (0.250 + 1.39i)2-s − 3-s + (−1.87 + 0.697i)4-s + 3.39i·5-s + (−0.250 − 1.39i)6-s + (−1.44 − 2.43i)8-s + 9-s + (−4.72 + 0.850i)10-s + 1.94i·11-s + (1.87 − 0.697i)12-s + 5.34i·13-s − 3.39i·15-s + (3.02 − 2.61i)16-s − 1.74i·17-s + (0.250 + 1.39i)18-s − 6.27·19-s + ⋯
L(s)  = 1  + (0.177 + 0.984i)2-s − 0.577·3-s + (−0.937 + 0.348i)4-s + 1.51i·5-s + (−0.102 − 0.568i)6-s + (−0.509 − 0.860i)8-s + 0.333·9-s + (−1.49 + 0.268i)10-s + 0.585i·11-s + (0.541 − 0.201i)12-s + 1.48i·13-s − 0.875i·15-s + (0.756 − 0.653i)16-s − 0.423i·17-s + (0.0590 + 0.328i)18-s − 1.43·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.712 + 0.701i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.712 + 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278790 - 0.680332i\)
\(L(\frac12)\) \(\approx\) \(0.278790 - 0.680332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.250 - 1.39i)T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 3.39iT - 5T^{2} \)
11 \( 1 - 1.94iT - 11T^{2} \)
13 \( 1 - 5.34iT - 13T^{2} \)
17 \( 1 + 1.74iT - 17T^{2} \)
19 \( 1 + 6.27T + 19T^{2} \)
23 \( 1 + 8.39iT - 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 + 4.59T + 31T^{2} \)
37 \( 1 + 1.78T + 37T^{2} \)
41 \( 1 - 1.32iT - 41T^{2} \)
43 \( 1 - 3.27iT - 43T^{2} \)
47 \( 1 - 8.22T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 - 0.507T + 59T^{2} \)
61 \( 1 + 7.63iT - 61T^{2} \)
67 \( 1 - 16.1iT - 67T^{2} \)
71 \( 1 - 1.11iT - 71T^{2} \)
73 \( 1 - 3.06iT - 73T^{2} \)
79 \( 1 - 6.77iT - 79T^{2} \)
83 \( 1 - 5.39T + 83T^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 - 6.46iT - 97T^{2} \)
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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

−11.11075195972977140440557597130, −10.39619812129822667229092556519, −9.485904379010991587472706000765, −8.454243496495802525237445196248, −7.24742361848453048794610383587, −6.66142097823627718807859536986, −6.23519116933049776146247797491, −4.76368506296978628713264725692, −3.98932257921873813007496949820, −2.44231312967821083392639401563, 0.42557544919503683521816132118, 1.65514511416910354814952332032, 3.40791202367358470281835003352, 4.48655893628995453309570693984, 5.37901165841224334409192308270, 5.95602264582610343035173356761, 7.83258348124875663352169115554, 8.608016465210628988948407902478, 9.340170748720623273290672998434, 10.38965109048796423018073290896

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