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Properties

Label 2-588-28.3-c1-0-25
Degree $2$
Conductor $588$
Sign $-0.184 + 0.982i$
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  + (−1.41 − 0.0865i)2-s + (−0.5 + 0.866i)3-s + (1.98 + 0.244i)4-s + (1.46 − 0.848i)5-s + (0.780 − 1.17i)6-s + (−2.78 − 0.516i)8-s + (−0.499 − 0.866i)9-s + (−2.14 + 1.06i)10-s + (−2.61 − 1.51i)11-s + (−1.20 + 1.59i)12-s − 6.04i·13-s + 1.69i·15-s + (3.88 + 0.970i)16-s + (−3.76 − 2.17i)17-s + (0.630 + 1.26i)18-s + (−0.561 − 0.972i)19-s + ⋯
L(s)  = 1  + (−0.998 − 0.0612i)2-s + (−0.288 + 0.499i)3-s + (0.992 + 0.122i)4-s + (0.656 − 0.379i)5-s + (0.318 − 0.481i)6-s + (−0.983 − 0.182i)8-s + (−0.166 − 0.288i)9-s + (−0.678 + 0.338i)10-s + (−0.788 − 0.455i)11-s + (−0.347 + 0.460i)12-s − 1.67i·13-s + 0.437i·15-s + (0.970 + 0.242i)16-s + (−0.912 − 0.526i)17-s + (0.148 + 0.298i)18-s + (−0.128 − 0.223i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.184 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.374060 - 0.450978i\)
\(L(\frac12)\) \(\approx\) \(0.374060 - 0.450978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0865i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.46 + 0.848i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.61 + 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.04iT - 13T^{2} \)
17 \( 1 + (3.76 + 2.17i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.561 + 0.972i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.61 - 1.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.56 - 6.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.73iT - 41T^{2} \)
43 \( 1 + 8.10iT - 43T^{2} \)
47 \( 1 + (5.12 + 8.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 + 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.16 + 4.71i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.79 + 1.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (2.93 + 1.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.08 + 2.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 + (-6.70 + 3.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.68iT - 97T^{2} \)
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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

−10.29740619406233943799056179418, −9.723909235125548252776367671336, −8.748842748008637598684010721139, −8.064865179120240229043065118814, −6.97226249596280683314955658518, −5.76725469241825901067549954723, −5.22761673798193398323164254414, −3.44371650407209748798240424396, −2.25174224134331134711094540979, −0.43859213532585830195383342022, 1.75439944079605989781360002372, 2.51163644004533420124433752018, 4.43691077322542032615066753975, 5.98210568301536252023190421883, 6.48942294468932148014042556957, 7.39031340822156288301466153650, 8.281751886041208126981636066463, 9.289987018052410625796613774891, 9.978891418527829064548736977890, 10.88608375966418209579674517899

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