Properties

Label 2-310464-1.1-c1-0-133
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  − 11-s − 5·17-s + 5·19-s + 5·23-s − 5·25-s + 3·29-s + 10·31-s + 3·37-s + 2·41-s − 43-s − 5·47-s + 4·53-s − 11·59-s − 2·61-s − 12·67-s + 11·71-s − 8·73-s + 8·79-s − 12·83-s + 8·89-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.301·11-s − 1.21·17-s + 1.14·19-s + 1.04·23-s − 25-s + 0.557·29-s + 1.79·31-s + 0.493·37-s + 0.312·41-s − 0.152·43-s − 0.729·47-s + 0.549·53-s − 1.43·59-s − 0.256·61-s − 1.46·67-s + 1.30·71-s − 0.936·73-s + 0.900·79-s − 1.31·83-s + 0.847·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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

\(L(1)\) \(\approx\) \(2.354951692\)
\(L(\frac12)\) \(\approx\) \(2.354951692\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.72491856858711, −12.05047023773667, −11.78459104876002, −11.30429538891643, −10.91154836594677, −10.30245391152569, −9.959330003835966, −9.453362984888132, −8.948815045555020, −8.610086837666718, −7.872361036417590, −7.697011248664969, −7.060387107321570, −6.510291040913064, −6.189363647376335, −5.569565870376043, −4.998212490531752, −4.543556854444919, −4.199587414748003, −3.306571553999772, −2.975206098184150, −2.422492095948001, −1.749646166137697, −1.066065839967173, −0.4449722053956590, 0.4449722053956590, 1.066065839967173, 1.749646166137697, 2.422492095948001, 2.975206098184150, 3.306571553999772, 4.199587414748003, 4.543556854444919, 4.998212490531752, 5.569565870376043, 6.189363647376335, 6.510291040913064, 7.060387107321570, 7.697011248664969, 7.872361036417590, 8.610086837666718, 8.948815045555020, 9.453362984888132, 9.959330003835966, 10.30245391152569, 10.91154836594677, 11.30429538891643, 11.78459104876002, 12.05047023773667, 12.72491856858711

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