Properties

Label 2-245-1.1-c5-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.6·2-s − 22.7·3-s + 81.4·4-s − 25·5-s + 242.·6-s − 527.·8-s + 273.·9-s + 266.·10-s + 521.·11-s − 1.85e3·12-s + 530.·13-s + 567.·15-s + 3.00e3·16-s + 1.99e3·17-s − 2.90e3·18-s + 2.49e3·19-s − 2.03e3·20-s − 5.55e3·22-s − 3.55e3·23-s + 1.19e4·24-s + 625·25-s − 5.65e3·26-s − 683.·27-s − 4.28e3·29-s − 6.05e3·30-s + 5.03e3·31-s − 1.51e4·32-s + ⋯
L(s)  = 1  − 1.88·2-s − 1.45·3-s + 2.54·4-s − 0.447·5-s + 2.74·6-s − 2.91·8-s + 1.12·9-s + 0.842·10-s + 1.29·11-s − 3.71·12-s + 0.870·13-s + 0.651·15-s + 2.93·16-s + 1.67·17-s − 2.11·18-s + 1.58·19-s − 1.13·20-s − 2.44·22-s − 1.40·23-s + 4.24·24-s + 0.200·25-s − 1.63·26-s − 0.180·27-s − 0.946·29-s − 1.22·30-s + 0.941·31-s − 2.62·32-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5199985229\)
\(L(\frac12)\) \(\approx\) \(0.5199985229\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.6T + 32T^{2} \)
3 \( 1 + 22.7T + 243T^{2} \)
11 \( 1 - 521.T + 1.61e5T^{2} \)
13 \( 1 - 530.T + 3.71e5T^{2} \)
17 \( 1 - 1.99e3T + 1.41e6T^{2} \)
19 \( 1 - 2.49e3T + 2.47e6T^{2} \)
23 \( 1 + 3.55e3T + 6.43e6T^{2} \)
29 \( 1 + 4.28e3T + 2.05e7T^{2} \)
31 \( 1 - 5.03e3T + 2.86e7T^{2} \)
37 \( 1 + 2.62e3T + 6.93e7T^{2} \)
41 \( 1 - 1.82e3T + 1.15e8T^{2} \)
43 \( 1 + 5.08e3T + 1.47e8T^{2} \)
47 \( 1 + 4.98e3T + 2.29e8T^{2} \)
53 \( 1 - 1.90e4T + 4.18e8T^{2} \)
59 \( 1 + 8.93e3T + 7.14e8T^{2} \)
61 \( 1 - 1.35e4T + 8.44e8T^{2} \)
67 \( 1 + 3.37e3T + 1.35e9T^{2} \)
71 \( 1 - 4.69e4T + 1.80e9T^{2} \)
73 \( 1 - 9.79e3T + 2.07e9T^{2} \)
79 \( 1 - 6.85e4T + 3.07e9T^{2} \)
83 \( 1 + 2.55e4T + 3.93e9T^{2} \)
89 \( 1 + 1.12e5T + 5.58e9T^{2} \)
97 \( 1 - 1.56e5T + 8.58e9T^{2} \)
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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

−11.20014895088847964437218967129, −10.16893879237515443993369326405, −9.523141077569923203464357446599, −8.295397614363897938816072889267, −7.38324057593702539078079870166, −6.41435932055632083811359868655, −5.60810969459520444745431334988, −3.56340642120597535605975324378, −1.38634450490750344577128333820, −0.69396773514021966860137750197, 0.69396773514021966860137750197, 1.38634450490750344577128333820, 3.56340642120597535605975324378, 5.60810969459520444745431334988, 6.41435932055632083811359868655, 7.38324057593702539078079870166, 8.295397614363897938816072889267, 9.523141077569923203464357446599, 10.16893879237515443993369326405, 11.20014895088847964437218967129

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