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Properties

Label 2-588-7.4-c1-0-6
Degree $2$
Conductor $588$
Sign $-0.991 + 0.126i$
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  + (0.5 − 0.866i)3-s + (−2 − 3.46i)5-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s − 6·13-s − 3.99·15-s + (2 − 3.46i)17-s + (2 + 3.46i)19-s + (−1 − 1.73i)23-s + (−5.49 + 9.52i)25-s − 0.999·27-s − 2·29-s + (0.999 + 1.73i)33-s + (−1 − 1.73i)37-s + (−3 + 5.19i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.894 − 1.54i)5-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s − 1.66·13-s − 1.03·15-s + (0.485 − 0.840i)17-s + (0.458 + 0.794i)19-s + (−0.208 − 0.361i)23-s + (−1.09 + 1.90i)25-s − 0.192·27-s − 0.371·29-s + (0.174 + 0.301i)33-s + (−0.164 − 0.284i)37-s + (−0.480 + 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0465567 - 0.733636i\)
\(L(\frac12)\) \(\approx\) \(0.0465567 - 0.733636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
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

−9.961088322041965499909387192052, −9.398573899880374312283994840274, −8.311258261901049103493566706408, −7.75080962545215201824874291030, −6.99796973659882363927068289391, −5.31326106700197727345927486524, −4.80124669537225884660156073464, −3.54736380962743112650401391679, −1.98607185152889478833389416745, −0.38039819799258426611661102810, 2.58193713024375482736455347095, 3.30307596880590792904690426257, 4.36919617828463971556055836802, 5.60826205639852589677879544516, 6.86767565388197332704314590226, 7.54300189904740346619875597384, 8.294942935677157110912298828858, 9.626486515160467124569968352516, 10.25274891919370468814817096230, 11.07443876182715986351125425186

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