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Properties

Label 2-588-7.4-c1-0-3
Degree $2$
Conductor $588$
Sign $0.605 + 0.795i$
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 + (−1 − 1.73i)5-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 3·13-s + 1.99·15-s + (4 − 6.92i)17-s + (−0.5 − 0.866i)19-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s + 0.999·27-s + 4·29-s + (1.5 − 2.59i)31-s + (−0.999 − 1.73i)33-s + (0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.832·13-s + 0.516·15-s + (0.970 − 1.68i)17-s + (−0.114 − 0.198i)19-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s + 0.192·27-s + 0.742·29-s + (0.269 − 0.466i)31-s + (−0.174 − 0.301i)33-s + (0.0821 + 0.142i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 588,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00696 - 0.499178i\)
\(L(\frac12)\) \(\approx\) \(1.00696 - 0.499178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.46581362455315820703270200404, −9.774325998618023450521222488917, −8.793829749877786594326541294749, −8.078314047019151047749279353773, −6.99022940396525788328374918806, −5.83681122472412503926110971329, −4.83450555496741839915063303491, −4.15554911104864310336645817503, −2.73572212186900775081370168887, −0.72500536789799164666846476553, 1.46196899898935132217137690139, 3.11440067935756924438277152789, 3.98277885183252021178572561827, 5.66878862298854949462179269936, 6.17381983040674805170905629738, 7.35687613609293183882966911755, 7.999192546505736906353541738593, 8.895836238776167463077657651039, 10.38655405434576491905132926398, 10.67474349052871886641334513973

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