Properties

Label 2-2450-1.1-c3-0-30
Degree $2$
Conductor $2450$
Sign $1$
Analytic cond. $144.554$
Root an. cond. $12.0230$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 4·4-s − 4·6-s − 8·8-s − 23·9-s − 28·11-s + 8·12-s + 12·13-s + 16·16-s − 64·17-s + 46·18-s + 60·19-s + 56·22-s + 58·23-s − 16·24-s − 24·26-s − 100·27-s + 90·29-s + 128·31-s − 32·32-s − 56·33-s + 128·34-s − 92·36-s − 236·37-s − 120·38-s + 24·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.384·3-s + 1/2·4-s − 0.272·6-s − 0.353·8-s − 0.851·9-s − 0.767·11-s + 0.192·12-s + 0.256·13-s + 1/4·16-s − 0.913·17-s + 0.602·18-s + 0.724·19-s + 0.542·22-s + 0.525·23-s − 0.136·24-s − 0.181·26-s − 0.712·27-s + 0.576·29-s + 0.741·31-s − 0.176·32-s − 0.295·33-s + 0.645·34-s − 0.425·36-s − 1.04·37-s − 0.512·38-s + 0.0985·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(144.554\)
Root analytic conductor: \(12.0230\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.113319948\)
\(L(\frac12)\) \(\approx\) \(1.113319948\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 2 T + p^{3} T^{2} \)
11 \( 1 + 28 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 + 64 T + p^{3} T^{2} \)
19 \( 1 - 60 T + p^{3} T^{2} \)
23 \( 1 - 58 T + p^{3} T^{2} \)
29 \( 1 - 90 T + p^{3} T^{2} \)
31 \( 1 - 128 T + p^{3} T^{2} \)
37 \( 1 + 236 T + p^{3} T^{2} \)
41 \( 1 + 242 T + p^{3} T^{2} \)
43 \( 1 + 362 T + p^{3} T^{2} \)
47 \( 1 - 226 T + p^{3} T^{2} \)
53 \( 1 - 108 T + p^{3} T^{2} \)
59 \( 1 - 20 T + p^{3} T^{2} \)
61 \( 1 + 542 T + p^{3} T^{2} \)
67 \( 1 - 434 T + p^{3} T^{2} \)
71 \( 1 + 1128 T + p^{3} T^{2} \)
73 \( 1 - 632 T + p^{3} T^{2} \)
79 \( 1 + 720 T + p^{3} T^{2} \)
83 \( 1 + 478 T + p^{3} T^{2} \)
89 \( 1 - 490 T + p^{3} T^{2} \)
97 \( 1 - 1456 T + p^{3} T^{2} \)
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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

−8.645726804087319293279850508035, −8.023816961345608917900614121835, −7.20543530056415563981151174857, −6.43084327006523472905530780945, −5.53158159446791779322289813210, −4.72084670890444112860475186871, −3.38184150303064431575487857062, −2.75326217726992453672745935728, −1.77875137616764951790212959227, −0.49273099076455869475803941641, 0.49273099076455869475803941641, 1.77875137616764951790212959227, 2.75326217726992453672745935728, 3.38184150303064431575487857062, 4.72084670890444112860475186871, 5.53158159446791779322289813210, 6.43084327006523472905530780945, 7.20543530056415563981151174857, 8.023816961345608917900614121835, 8.645726804087319293279850508035

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