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Properties

Label 2-588-7.5-c2-0-11
Degree $2$
Conductor $588$
Sign $-0.0956 + 0.995i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)3-s + (0.804 + 0.464i)5-s + (1.5 − 2.59i)9-s + (−4.84 − 8.39i)11-s − 15.9i·13-s + 1.60·15-s + (−9.14 + 5.27i)17-s + (−6.25 − 3.61i)19-s + (5.65 − 9.78i)23-s + (−12.0 − 20.9i)25-s − 5.19i·27-s + 46.3·29-s + (0.418 − 0.241i)31-s + (−14.5 − 8.39i)33-s + (−1.24 + 2.14i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.160 + 0.0929i)5-s + (0.166 − 0.288i)9-s + (−0.440 − 0.762i)11-s − 1.22i·13-s + 0.107·15-s + (−0.537 + 0.310i)17-s + (−0.329 − 0.190i)19-s + (0.245 − 0.425i)23-s + (−0.482 − 0.836i)25-s − 0.192i·27-s + 1.59·29-s + (0.0134 − 0.00779i)31-s + (−0.440 − 0.254i)33-s + (−0.0335 + 0.0580i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0956 + 0.995i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.0956 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.771146760\)
\(L(\frac12)\) \(\approx\) \(1.771146760\)
\(L(2)\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-0.804 - 0.464i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (4.84 + 8.39i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 15.9iT - 169T^{2} \)
17 \( 1 + (9.14 - 5.27i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.25 + 3.61i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-5.65 + 9.78i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 46.3T + 841T^{2} \)
31 \( 1 + (-0.418 + 0.241i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (1.24 - 2.14i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 55.8iT - 1.68e3T^{2} \)
43 \( 1 - 60.6T + 1.84e3T^{2} \)
47 \( 1 + (31.6 + 18.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-14.2 - 24.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (81.4 - 47.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (95.4 + 55.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (41.0 + 71.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 127.T + 5.04e3T^{2} \)
73 \( 1 + (-40.0 + 23.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (9.35 - 16.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 59.6iT - 6.88e3T^{2} \)
89 \( 1 + (-61.5 - 35.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 102. iT - 9.40e3T^{2} \)
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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

−10.46760078581081913500501242250, −9.256109099238678828101342037813, −8.377565755554915257202118376955, −7.81344630484141600522744944095, −6.60245457798695281059863854923, −5.79767289120527673170147743781, −4.57610180071451838712933220375, −3.26054778011134148456717249602, −2.34823753316247446869997875873, −0.61888852623346110427723901108, 1.68799010100048453265580169253, 2.83340599893290762755429023713, 4.21706967739013304341326675499, 4.91937915936334211783892425562, 6.27989280755993633760123812513, 7.20981518855079096999721378359, 8.119074328221839082652339181206, 9.161554063473552474894001578194, 9.648340225421362309535382592984, 10.63874237499715577334685897789

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