Properties

Label 2-245-1.1-c5-0-31
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $39.2940$
Root an. cond. $6.26849$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.350·2-s − 21.5·3-s − 31.8·4-s − 25·5-s − 7.56·6-s − 22.3·8-s + 223.·9-s − 8.76·10-s − 193.·11-s + 688.·12-s + 691.·13-s + 539.·15-s + 1.01e3·16-s + 649.·17-s + 78.2·18-s + 696.·19-s + 796.·20-s − 67.7·22-s − 2.97e3·23-s + 483.·24-s + 625·25-s + 242.·26-s + 428.·27-s + 6.77e3·29-s + 189.·30-s + 151.·31-s + 1.07e3·32-s + ⋯
L(s)  = 1  + 0.0619·2-s − 1.38·3-s − 0.996·4-s − 0.447·5-s − 0.0858·6-s − 0.123·8-s + 0.918·9-s − 0.0277·10-s − 0.481·11-s + 1.37·12-s + 1.13·13-s + 0.619·15-s + 0.988·16-s + 0.545·17-s + 0.0569·18-s + 0.442·19-s + 0.445·20-s − 0.0298·22-s − 1.17·23-s + 0.171·24-s + 0.200·25-s + 0.0702·26-s + 0.113·27-s + 1.49·29-s + 0.0383·30-s + 0.0283·31-s + 0.184·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(39.2940\)
Root analytic conductor: \(6.26849\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
7 \( 1 \)
good2 \( 1 - 0.350T + 32T^{2} \)
3 \( 1 + 21.5T + 243T^{2} \)
11 \( 1 + 193.T + 1.61e5T^{2} \)
13 \( 1 - 691.T + 3.71e5T^{2} \)
17 \( 1 - 649.T + 1.41e6T^{2} \)
19 \( 1 - 696.T + 2.47e6T^{2} \)
23 \( 1 + 2.97e3T + 6.43e6T^{2} \)
29 \( 1 - 6.77e3T + 2.05e7T^{2} \)
31 \( 1 - 151.T + 2.86e7T^{2} \)
37 \( 1 + 1.10e4T + 6.93e7T^{2} \)
41 \( 1 - 7.36e3T + 1.15e8T^{2} \)
43 \( 1 + 1.92e4T + 1.47e8T^{2} \)
47 \( 1 - 1.27e4T + 2.29e8T^{2} \)
53 \( 1 - 9.07e3T + 4.18e8T^{2} \)
59 \( 1 + 3.69e4T + 7.14e8T^{2} \)
61 \( 1 + 2.34e4T + 8.44e8T^{2} \)
67 \( 1 - 6.44e4T + 1.35e9T^{2} \)
71 \( 1 + 8.31e4T + 1.80e9T^{2} \)
73 \( 1 - 2.89e4T + 2.07e9T^{2} \)
79 \( 1 - 1.69e3T + 3.07e9T^{2} \)
83 \( 1 - 8.03e4T + 3.93e9T^{2} \)
89 \( 1 - 1.18e5T + 5.58e9T^{2} \)
97 \( 1 + 1.47e5T + 8.58e9T^{2} \)
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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

−10.72011245779481724040637914562, −10.08665414621770953583725324247, −8.741877664867371947082940199638, −7.84202419990833750371016030838, −6.40010112330960362476884844430, −5.51552553048396383021758408746, −4.64021937632334924522700308637, −3.48053194199081156568177165902, −1.05822187658478752204260619945, 0, 1.05822187658478752204260619945, 3.48053194199081156568177165902, 4.64021937632334924522700308637, 5.51552553048396383021758408746, 6.40010112330960362476884844430, 7.84202419990833750371016030838, 8.741877664867371947082940199638, 10.08665414621770953583725324247, 10.72011245779481724040637914562

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