Properties

Label 2-8280-69.68-c1-0-90
Degree $2$
Conductor $8280$
Sign $-0.993 + 0.112i$
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 − 2.53i·7-s + 2.07·11-s − 3.49·13-s + 0.496·17-s − 5.97i·19-s + (4.20 + 2.31i)23-s + 25-s − 3.28i·29-s − 5.25·31-s + 2.53i·35-s − 7.40i·37-s + 8.94i·41-s + 9.51i·43-s − 10.9i·47-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.957i·7-s + 0.624·11-s − 0.968·13-s + 0.120·17-s − 1.37i·19-s + (0.876 + 0.481i)23-s + 0.200·25-s − 0.610i·29-s − 0.944·31-s + 0.428i·35-s − 1.21i·37-s + 1.39i·41-s + 1.45i·43-s − 1.59i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.993 + 0.112i$
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: \((2,\ 8280,\ (\ :1/2),\ -0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6885555821\)
\(L(\frac12)\) \(\approx\) \(0.6885555821\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.20 - 2.31i)T \)
good7 \( 1 + 2.53iT - 7T^{2} \)
11 \( 1 - 2.07T + 11T^{2} \)
13 \( 1 + 3.49T + 13T^{2} \)
17 \( 1 - 0.496T + 17T^{2} \)
19 \( 1 + 5.97iT - 19T^{2} \)
29 \( 1 + 3.28iT - 29T^{2} \)
31 \( 1 + 5.25T + 31T^{2} \)
37 \( 1 + 7.40iT - 37T^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 - 9.51iT - 43T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + 7.85T + 53T^{2} \)
59 \( 1 + 6.38iT - 59T^{2} \)
61 \( 1 - 2.30iT - 61T^{2} \)
67 \( 1 + 4.82iT - 67T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 - 0.143T + 73T^{2} \)
79 \( 1 - 1.88iT - 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 - 9.17iT - 97T^{2} \)
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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

−7.51454018289671833311019562559, −6.86873404750395500155030702085, −6.31385473547707380615164563668, −5.13242559797558684043147185977, −4.71067896248104396366233596023, −3.88970228136405603727787646112, −3.22687527439308369876701519935, −2.26998044564580183090085287662, −1.11458487442881172639392788656, −0.17548451103684557756858208127, 1.29326689100452086478841001105, 2.23857047990942967642513382585, 3.08053991400123008806738584016, 3.83696940312973486497794195211, 4.67123522762738649922156165014, 5.40988625619688739303276783622, 5.97200088159587947553693876931, 6.89125464140321197052960250739, 7.37577415371449419277849104992, 8.192421292193900556644004388112

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