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Properties

Label 2-588-7.2-c7-0-0
Degree $2$
Conductor $588$
Sign $-0.266 + 0.963i$
Analytic cond. $183.682$
Root an. cond. $13.5529$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 + 23.3i)3-s + (−120 + 207. i)5-s + (−364.5 + 631. i)9-s + (−351 − 607. i)11-s + 3.95e3·13-s − 6.48e3·15-s + (−1.70e3 − 2.95e3i)17-s + (−2.45e4 + 4.24e4i)19-s + (5.75e3 − 9.97e3i)23-s + (1.02e4 + 1.77e4i)25-s − 1.96e4·27-s + 4.96e4·29-s + (−5.66e4 − 9.81e4i)31-s + (9.47e3 − 1.64e4i)33-s + (3.34e4 − 5.79e4i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.429 + 0.743i)5-s + (−0.166 + 0.288i)9-s + (−0.0795 − 0.137i)11-s + 0.499·13-s − 0.495·15-s + (−0.0841 − 0.145i)17-s + (−0.820 + 1.42i)19-s + (0.0986 − 0.170i)23-s + (0.131 + 0.227i)25-s − 0.192·27-s + 0.378·29-s + (−0.341 − 0.591i)31-s + (0.0459 − 0.0795i)33-s + (0.108 − 0.188i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(183.682\)
Root analytic conductor: \(13.5529\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :7/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.08056292907\)
\(L(\frac12)\) \(\approx\) \(0.08056292907\)
\(L(\frac{9}{2})\) 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 \( 1 + (-13.5 - 23.3i)T \)
7 \( 1 \)
good5 \( 1 + (120 - 207. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (351 + 607. i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 - 3.95e3T + 6.27e7T^{2} \)
17 \( 1 + (1.70e3 + 2.95e3i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (2.45e4 - 4.24e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (-5.75e3 + 9.97e3i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 - 4.96e4T + 1.72e10T^{2} \)
31 \( 1 + (5.66e4 + 9.81e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (-3.34e4 + 5.79e4i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 - 3.60e5T + 1.94e11T^{2} \)
43 \( 1 + 7.65e5T + 2.71e11T^{2} \)
47 \( 1 + (6.72e5 - 1.16e6i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (1.79e5 + 3.10e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-4.65e5 - 8.05e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (6.59e5 - 1.14e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (9.46e5 + 1.63e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 2.27e5T + 9.09e12T^{2} \)
73 \( 1 + (-3.92e5 - 6.79e5i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-1.05e6 + 1.81e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + 8.62e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.95e6 + 5.11e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 7.73e5T + 8.07e13T^{2} \)
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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

−10.28565362324233627190455855973, −9.353823132220563114240320320678, −8.368760544321691989042565484949, −7.69380658638387869247095417606, −6.60056421081598846498107604421, −5.74609149205146222878056715020, −4.47470294099587718420934114880, −3.63061136955871261108282482774, −2.80271193523160402905173852618, −1.55018273444343400705910321363, 0.01572704086729964171158144476, 0.940644247256849907555213777761, 2.02827724711382770840567649591, 3.18619571830922931356354109628, 4.32460038403474607941705563049, 5.15954334004229563405030312439, 6.42402888367121847406509468878, 7.14363531953807893442976682750, 8.345724912714822295177923069242, 8.631132802688597421831654762873

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