Properties

Label 2-245-1.1-c5-0-2
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  − 6.05·2-s − 15.9·3-s + 4.62·4-s − 25·5-s + 96.6·6-s + 165.·8-s + 12.2·9-s + 151.·10-s + 708.·11-s − 73.8·12-s − 1.12e3·13-s + 399.·15-s − 1.15e3·16-s + 393.·17-s − 73.9·18-s − 1.79e3·19-s − 115.·20-s − 4.28e3·22-s − 3.57e3·23-s − 2.64e3·24-s + 625·25-s + 6.80e3·26-s + 3.68e3·27-s − 2.14·29-s − 2.41e3·30-s − 8.77e3·31-s + 1.66e3·32-s + ⋯
L(s)  = 1  − 1.06·2-s − 1.02·3-s + 0.144·4-s − 0.447·5-s + 1.09·6-s + 0.915·8-s + 0.0502·9-s + 0.478·10-s + 1.76·11-s − 0.148·12-s − 1.84·13-s + 0.458·15-s − 1.12·16-s + 0.330·17-s − 0.0537·18-s − 1.13·19-s − 0.0646·20-s − 1.88·22-s − 1.41·23-s − 0.937·24-s + 0.200·25-s + 1.97·26-s + 0.973·27-s − 0.000474·29-s − 0.490·30-s − 1.63·31-s + 0.286·32-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

\(L(3)\) \(\approx\) \(0.2290440368\)
\(L(\frac12)\) \(\approx\) \(0.2290440368\)
\(L(\frac{7}{2})\) 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 \( 1 + 6.05T + 32T^{2} \)
3 \( 1 + 15.9T + 243T^{2} \)
11 \( 1 - 708.T + 1.61e5T^{2} \)
13 \( 1 + 1.12e3T + 3.71e5T^{2} \)
17 \( 1 - 393.T + 1.41e6T^{2} \)
19 \( 1 + 1.79e3T + 2.47e6T^{2} \)
23 \( 1 + 3.57e3T + 6.43e6T^{2} \)
29 \( 1 + 2.14T + 2.05e7T^{2} \)
31 \( 1 + 8.77e3T + 2.86e7T^{2} \)
37 \( 1 + 4.36e3T + 6.93e7T^{2} \)
41 \( 1 - 8.41e3T + 1.15e8T^{2} \)
43 \( 1 + 1.81e4T + 1.47e8T^{2} \)
47 \( 1 + 8.20e3T + 2.29e8T^{2} \)
53 \( 1 + 1.43e4T + 4.18e8T^{2} \)
59 \( 1 - 1.01e4T + 7.14e8T^{2} \)
61 \( 1 - 2.87e3T + 8.44e8T^{2} \)
67 \( 1 - 3.73e4T + 1.35e9T^{2} \)
71 \( 1 + 3.25e3T + 1.80e9T^{2} \)
73 \( 1 + 6.12e3T + 2.07e9T^{2} \)
79 \( 1 - 7.54e3T + 3.07e9T^{2} \)
83 \( 1 + 1.97e3T + 3.93e9T^{2} \)
89 \( 1 - 4.68e4T + 5.58e9T^{2} \)
97 \( 1 + 1.97e4T + 8.58e9T^{2} \)
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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

−11.18516614800471445451045682832, −10.16554462918299175729688786282, −9.391996738530907213453279125586, −8.405497174411177911658556553578, −7.30686630170569941105557147677, −6.41761379473763646029645578164, −5.03214460040245531016614314070, −3.99329301763273863964942439298, −1.81881897729783519785762245566, −0.34625618278816309167364647218, 0.34625618278816309167364647218, 1.81881897729783519785762245566, 3.99329301763273863964942439298, 5.03214460040245531016614314070, 6.41761379473763646029645578164, 7.30686630170569941105557147677, 8.405497174411177911658556553578, 9.391996738530907213453279125586, 10.16554462918299175729688786282, 11.18516614800471445451045682832

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