Properties

Label 2-310464-1.1-c1-0-127
Degree $2$
Conductor $310464$
Sign $1$
Analytic cond. $2479.06$
Root an. cond. $49.7902$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 11-s + 13-s + 4·17-s − 3·19-s + 6·23-s − 4·25-s + 7·29-s + 4·31-s − 37-s − 4·41-s − 2·43-s − 7·47-s + 10·53-s − 55-s − 9·59-s − 2·61-s + 65-s − 9·67-s − 4·71-s + 9·73-s − 16·79-s + 4·83-s + 4·85-s − 2·89-s − 3·95-s + 14·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.301·11-s + 0.277·13-s + 0.970·17-s − 0.688·19-s + 1.25·23-s − 4/5·25-s + 1.29·29-s + 0.718·31-s − 0.164·37-s − 0.624·41-s − 0.304·43-s − 1.02·47-s + 1.37·53-s − 0.134·55-s − 1.17·59-s − 0.256·61-s + 0.124·65-s − 1.09·67-s − 0.474·71-s + 1.05·73-s − 1.80·79-s + 0.439·83-s + 0.433·85-s − 0.211·89-s − 0.307·95-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310464\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(2479.06\)
Root analytic conductor: \(49.7902\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 310464,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.544962030\)
\(L(\frac12)\) \(\approx\) \(2.544962030\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.70337259236620, −12.17097356681721, −11.75599500660113, −11.38740278282631, −10.58517156254679, −10.46776345704467, −9.967312256770414, −9.512422979642966, −8.948021434592188, −8.492083552386096, −8.109886152524176, −7.577617067893115, −7.017392470412496, −6.544576226291633, −6.079152101766936, −5.627862382822709, −5.024377846128510, −4.676036889771961, −4.038849232042978, −3.384887977002076, −2.913225905781362, −2.445370898543400, −1.624664311157592, −1.230696069186565, −0.4306372779603013, 0.4306372779603013, 1.230696069186565, 1.624664311157592, 2.445370898543400, 2.913225905781362, 3.384887977002076, 4.038849232042978, 4.676036889771961, 5.024377846128510, 5.627862382822709, 6.079152101766936, 6.544576226291633, 7.017392470412496, 7.577617067893115, 8.109886152524176, 8.492083552386096, 8.948021434592188, 9.512422979642966, 9.967312256770414, 10.46776345704467, 10.58517156254679, 11.38740278282631, 11.75599500660113, 12.17097356681721, 12.70337259236620

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