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Properties

Label 2-588-12.11-c1-0-68
Degree $2$
Conductor $588$
Sign $-0.910 + 0.414i$
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  + (1.29 − 0.559i)2-s + (−1.60 − 0.653i)3-s + (1.37 − 1.45i)4-s − 3.97i·5-s + (−2.44 + 0.0487i)6-s + (0.969 − 2.65i)8-s + (2.14 + 2.09i)9-s + (−2.22 − 5.15i)10-s − 3.30·11-s + (−3.15 + 1.43i)12-s + 0.850·13-s + (−2.59 + 6.37i)15-s + (−0.228 − 3.99i)16-s + 0.576i·17-s + (3.95 + 1.52i)18-s + 3.59i·19-s + ⋯
L(s)  = 1  + (0.918 − 0.395i)2-s + (−0.926 − 0.377i)3-s + (0.686 − 0.726i)4-s − 1.77i·5-s + (−0.999 + 0.0199i)6-s + (0.342 − 0.939i)8-s + (0.715 + 0.699i)9-s + (−0.703 − 1.63i)10-s − 0.996·11-s + (−0.910 + 0.414i)12-s + 0.235·13-s + (−0.670 + 1.64i)15-s + (−0.0570 − 0.998i)16-s + 0.139i·17-s + (0.933 + 0.358i)18-s + 0.823i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.910 + 0.414i$
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: \((2,\ 588,\ (\ :1/2),\ -0.910 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356569 - 1.64515i\)
\(L(\frac12)\) \(\approx\) \(0.356569 - 1.64515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 0.559i)T \)
3 \( 1 + (1.60 + 0.653i)T \)
7 \( 1 \)
good5 \( 1 + 3.97iT - 5T^{2} \)
11 \( 1 + 3.30T + 11T^{2} \)
13 \( 1 - 0.850T + 13T^{2} \)
17 \( 1 - 0.576iT - 17T^{2} \)
19 \( 1 - 3.59iT - 19T^{2} \)
23 \( 1 - 3.96T + 23T^{2} \)
29 \( 1 + 3.37iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 4.29T + 37T^{2} \)
41 \( 1 + 5.35iT - 41T^{2} \)
43 \( 1 + 7.66iT - 43T^{2} \)
47 \( 1 + 9.09T + 47T^{2} \)
53 \( 1 - 0.608iT - 53T^{2} \)
59 \( 1 - 7.87T + 59T^{2} \)
61 \( 1 - 9.74T + 61T^{2} \)
67 \( 1 + 9.04iT - 67T^{2} \)
71 \( 1 - 9.15T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 - 4.13T + 83T^{2} \)
89 \( 1 + 5.75iT - 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−10.52604378630863070959046408521, −9.731236259680045955280197807984, −8.479236325245404527478317091632, −7.53739070542758517910289482848, −6.30501266063859822833151037582, −5.31205419418284317357655663760, −5.01313613333456824275696423044, −3.88694214304201082218352526760, −1.99393406976163929265279931342, −0.791298490640943696436118898623, 2.57935722072271030332345056854, 3.45614461259482342446070531523, 4.68068636855112214031284643478, 5.63641602957630996501009387688, 6.53819361816600406161532005489, 7.05864167273168176585280605354, 7.998935700056708782040634649162, 9.667125955500813746168344427164, 10.58463384468769554141932577021, 11.21572737762576940926000674158

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