Properties

Label 2-8280-1.1-c1-0-36
Degree $2$
Conductor $8280$
Sign $1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3.44·7-s + 2.44·11-s − 4.44·13-s + 0.550·17-s + 2.44·19-s + 23-s + 25-s + 7.89·29-s − 7.89·31-s − 3.44·35-s + 8.34·37-s + 1.89·41-s − 0.898·43-s + 1.55·47-s + 4.89·49-s − 7.44·53-s − 2.44·55-s + 59-s + 4.44·61-s + 4.44·65-s − 9.44·67-s + 5.89·71-s − 3.55·73-s + 8.44·77-s + 15.4·83-s − 0.550·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.30·7-s + 0.738·11-s − 1.23·13-s + 0.133·17-s + 0.561·19-s + 0.208·23-s + 0.200·25-s + 1.46·29-s − 1.41·31-s − 0.583·35-s + 1.37·37-s + 0.296·41-s − 0.137·43-s + 0.226·47-s + 0.699·49-s − 1.02·53-s − 0.330·55-s + 0.130·59-s + 0.569·61-s + 0.551·65-s − 1.15·67-s + 0.700·71-s − 0.415·73-s + 0.962·77-s + 1.69·83-s − 0.0597·85-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.280710849\)
\(L(\frac12)\) \(\approx\) \(2.280710849\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good7 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 - 0.550T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
29 \( 1 - 7.89T + 29T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 - 8.34T + 37T^{2} \)
41 \( 1 - 1.89T + 41T^{2} \)
43 \( 1 + 0.898T + 43T^{2} \)
47 \( 1 - 1.55T + 47T^{2} \)
53 \( 1 + 7.44T + 53T^{2} \)
59 \( 1 - T + 59T^{2} \)
61 \( 1 - 4.44T + 61T^{2} \)
67 \( 1 + 9.44T + 67T^{2} \)
71 \( 1 - 5.89T + 71T^{2} \)
73 \( 1 + 3.55T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 8.89T + 89T^{2} \)
97 \( 1 + 1.10T + 97T^{2} \)
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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

−7.75857196629752521578968267791, −7.29966232800536056885740354701, −6.52801297537257888775635143058, −5.61548937801420973302995396939, −4.83062437860869042103286902910, −4.48734213369371259375248458292, −3.54268788289314143687549297091, −2.61870125683197535643590804398, −1.71767752383627765032714147744, −0.77211104447975082383245136030, 0.77211104447975082383245136030, 1.71767752383627765032714147744, 2.61870125683197535643590804398, 3.54268788289314143687549297091, 4.48734213369371259375248458292, 4.83062437860869042103286902910, 5.61548937801420973302995396939, 6.52801297537257888775635143058, 7.29966232800536056885740354701, 7.75857196629752521578968267791

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