Properties

Label 2-8280-69.68-c1-0-21
Degree $2$
Conductor $8280$
Sign $-0.0887 - 0.996i$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
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  − 5-s + 2.84i·7-s − 2.95·11-s − 0.451·13-s + 4.38·17-s − 7.84i·19-s + (3.10 + 3.65i)23-s + 25-s + 3.01i·29-s − 6.09·31-s − 2.84i·35-s − 0.619i·37-s − 5.35i·41-s + 3.75i·43-s + 5.23i·47-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.07i·7-s − 0.891·11-s − 0.125·13-s + 1.06·17-s − 1.80i·19-s + (0.647 + 0.762i)23-s + 0.200·25-s + 0.560i·29-s − 1.09·31-s − 0.481i·35-s − 0.101i·37-s − 0.836i·41-s + 0.572i·43-s + 0.763i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0887 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0887 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.0887 - 0.996i$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8280} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ -0.0887 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.276209431\)
\(L(\frac12)\) \(\approx\) \(1.276209431\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
23 \( 1 + (-3.10 - 3.65i)T \)
good7 \( 1 - 2.84iT - 7T^{2} \)
11 \( 1 + 2.95T + 11T^{2} \)
13 \( 1 + 0.451T + 13T^{2} \)
17 \( 1 - 4.38T + 17T^{2} \)
19 \( 1 + 7.84iT - 19T^{2} \)
29 \( 1 - 3.01iT - 29T^{2} \)
31 \( 1 + 6.09T + 31T^{2} \)
37 \( 1 + 0.619iT - 37T^{2} \)
41 \( 1 + 5.35iT - 41T^{2} \)
43 \( 1 - 3.75iT - 43T^{2} \)
47 \( 1 - 5.23iT - 47T^{2} \)
53 \( 1 - 14.0T + 53T^{2} \)
59 \( 1 + 11.8iT - 59T^{2} \)
61 \( 1 - 9.81iT - 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 - 1.07T + 73T^{2} \)
79 \( 1 - 8.71iT - 79T^{2} \)
83 \( 1 - 3.20T + 83T^{2} \)
89 \( 1 + 3.64T + 89T^{2} \)
97 \( 1 - 8.46iT - 97T^{2} \)
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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

−7.909155653289915160154184072473, −7.34129016978650373726845660698, −6.76907194302992188031854428631, −5.57150870045104864175744606791, −5.40902982809934627267988774189, −4.61750883030199761970000008248, −3.55066300840965922814637877244, −2.85786185624484312052946918044, −2.20310786823272703208662816196, −0.895570293787353494986069769893, 0.37304532714953570124214991847, 1.36686570914359103901334244816, 2.47958464088409696827767342096, 3.51119956483186872410287252511, 3.89715098959799497826825203335, 4.81419564638070618652969153875, 5.52151880804564893420129250235, 6.21694099725815739835632897033, 7.26332163153325562735160602613, 7.49476482794833337003958810441

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