Properties

Label 2-245-1.1-c3-0-21
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
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-s + 8·3-s − 7·4-s + 5·5-s + 8·6-s − 15·8-s + 37·9-s + 5·10-s + 12·11-s − 56·12-s + 78·13-s + 40·15-s + 41·16-s + 94·17-s + 37·18-s − 40·19-s − 35·20-s + 12·22-s + 32·23-s − 120·24-s + 25·25-s + 78·26-s + 80·27-s − 50·29-s + 40·30-s + 248·31-s + 161·32-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.53·3-s − 7/8·4-s + 0.447·5-s + 0.544·6-s − 0.662·8-s + 1.37·9-s + 0.158·10-s + 0.328·11-s − 1.34·12-s + 1.66·13-s + 0.688·15-s + 0.640·16-s + 1.34·17-s + 0.484·18-s − 0.482·19-s − 0.391·20-s + 0.116·22-s + 0.290·23-s − 1.02·24-s + 1/5·25-s + 0.588·26-s + 0.570·27-s − 0.320·29-s + 0.243·30-s + 1.43·31-s + 0.889·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.361602356\)
\(L(\frac12)\) \(\approx\) \(3.361602356\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - p T \)
7 \( 1 \)
good2 \( 1 - T + p^{3} T^{2} \)
3 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 - 6 p T + p^{3} T^{2} \)
17 \( 1 - 94 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
23 \( 1 - 32 T + p^{3} T^{2} \)
29 \( 1 + 50 T + p^{3} T^{2} \)
31 \( 1 - 8 p T + p^{3} T^{2} \)
37 \( 1 + 434 T + p^{3} T^{2} \)
41 \( 1 + 402 T + p^{3} T^{2} \)
43 \( 1 + 68 T + p^{3} T^{2} \)
47 \( 1 + 536 T + p^{3} T^{2} \)
53 \( 1 - 22 T + p^{3} T^{2} \)
59 \( 1 - 560 T + p^{3} T^{2} \)
61 \( 1 - 278 T + p^{3} T^{2} \)
67 \( 1 + 164 T + p^{3} T^{2} \)
71 \( 1 - 672 T + p^{3} T^{2} \)
73 \( 1 + 82 T + p^{3} T^{2} \)
79 \( 1 + 1000 T + p^{3} T^{2} \)
83 \( 1 - 448 T + p^{3} T^{2} \)
89 \( 1 - 870 T + p^{3} T^{2} \)
97 \( 1 + 1026 T + p^{3} T^{2} \)
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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

−11.91152298250087572989483998083, −10.32449259698026339247485307302, −9.544242905357488317009195169613, −8.554194504948999398194468424229, −8.244769374928595340441659614417, −6.59879768933006076589824751827, −5.30167294480111430917057451079, −3.85010069753357916697674857257, −3.18707343436852699114006984956, −1.43668475600251259916328283691, 1.43668475600251259916328283691, 3.18707343436852699114006984956, 3.85010069753357916697674857257, 5.30167294480111430917057451079, 6.59879768933006076589824751827, 8.244769374928595340441659614417, 8.554194504948999398194468424229, 9.544242905357488317009195169613, 10.32449259698026339247485307302, 11.91152298250087572989483998083

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