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Properties

Label 2-588-21.20-c5-0-15
Degree $2$
Conductor $588$
Sign $-0.488 + 0.872i$
Analytic cond. $94.3056$
Root an. cond. $9.71111$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 + 15.5i)3-s − 69.8·5-s + (−239. − 43.2i)9-s + 465. i·11-s + 285. i·13-s + (97.2 − 1.08e3i)15-s + 898.·17-s + 2.09e3i·19-s + 4.90e3i·23-s + 1.76e3·25-s + (1.00e3 − 3.65e3i)27-s + 5.87e3i·29-s + 3.85e3i·31-s + (−7.23e3 − 648. i)33-s − 3.61e3·37-s + ⋯
L(s)  = 1  + (−0.0892 + 0.996i)3-s − 1.25·5-s + (−0.984 − 0.177i)9-s + 1.16i·11-s + 0.469i·13-s + (0.111 − 1.24i)15-s + 0.753·17-s + 1.33i·19-s + 1.93i·23-s + 0.563·25-s + (0.264 − 0.964i)27-s + 1.29i·29-s + 0.720i·31-s + (−1.15 − 0.103i)33-s − 0.434·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.488 + 0.872i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ -0.488 + 0.872i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8909922496\)
\(L(\frac12)\) \(\approx\) \(0.8909922496\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.39 - 15.5i)T \)
7 \( 1 \)
good5 \( 1 + 69.8T + 3.12e3T^{2} \)
11 \( 1 - 465. iT - 1.61e5T^{2} \)
13 \( 1 - 285. iT - 3.71e5T^{2} \)
17 \( 1 - 898.T + 1.41e6T^{2} \)
19 \( 1 - 2.09e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.90e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.87e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.85e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.61e3T + 6.93e7T^{2} \)
41 \( 1 - 1.71e4T + 1.15e8T^{2} \)
43 \( 1 + 1.85e4T + 1.47e8T^{2} \)
47 \( 1 - 2.42e3T + 2.29e8T^{2} \)
53 \( 1 - 7.61e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.06e4T + 7.14e8T^{2} \)
61 \( 1 - 1.56e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.97e3T + 1.35e9T^{2} \)
71 \( 1 - 1.62e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.46e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.61e4T + 3.07e9T^{2} \)
83 \( 1 - 2.74e3T + 3.93e9T^{2} \)
89 \( 1 - 1.01e5T + 5.58e9T^{2} \)
97 \( 1 + 2.34e4iT - 8.58e9T^{2} \)
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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

−10.39512350896279161197536456910, −9.733150763832826412061566985746, −8.828478928837068092911129909519, −7.82544645233373747552826923453, −7.16569626793204863244131868543, −5.73802115678882106231296564021, −4.80765127272082477422119370976, −3.89265128049339003387854445969, −3.26832918691912257069371712532, −1.51040807852404602037953738804, 0.31006399939765151973950312930, 0.70588602950228795784702917118, 2.48874162263175791447334564673, 3.39241383492159490719642501602, 4.62257019196225831524784769352, 5.85136091702490945336723380382, 6.68355823184572880381317824421, 7.72396213987259566361029267738, 8.181350894227954265885153620129, 8.992134806611633503920996494205

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