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Properties

Label 2-245-1.1-c5-0-56
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  + 9.75·2-s − 23.6·3-s + 63.0·4-s − 25·5-s − 230.·6-s + 303.·8-s + 315.·9-s − 243.·10-s + 160.·11-s − 1.49e3·12-s + 1.07e3·13-s + 590.·15-s + 937.·16-s − 1.63e3·17-s + 3.07e3·18-s − 2.05e3·19-s − 1.57e3·20-s + 1.56e3·22-s − 4.84e3·23-s − 7.16e3·24-s + 625·25-s + 1.05e4·26-s − 1.71e3·27-s − 4.20e3·29-s + 5.76e3·30-s − 8.39e3·31-s − 557.·32-s + ⋯
L(s)  = 1  + 1.72·2-s − 1.51·3-s + 1.97·4-s − 0.447·5-s − 2.61·6-s + 1.67·8-s + 1.29·9-s − 0.770·10-s + 0.399·11-s − 2.98·12-s + 1.77·13-s + 0.678·15-s + 0.915·16-s − 1.37·17-s + 2.23·18-s − 1.30·19-s − 0.881·20-s + 0.687·22-s − 1.90·23-s − 2.53·24-s + 0.200·25-s + 3.05·26-s − 0.453·27-s − 0.928·29-s + 1.16·30-s − 1.56·31-s − 0.0962·32-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 \( 1 - 9.75T + 32T^{2} \)
3 \( 1 + 23.6T + 243T^{2} \)
11 \( 1 - 160.T + 1.61e5T^{2} \)
13 \( 1 - 1.07e3T + 3.71e5T^{2} \)
17 \( 1 + 1.63e3T + 1.41e6T^{2} \)
19 \( 1 + 2.05e3T + 2.47e6T^{2} \)
23 \( 1 + 4.84e3T + 6.43e6T^{2} \)
29 \( 1 + 4.20e3T + 2.05e7T^{2} \)
31 \( 1 + 8.39e3T + 2.86e7T^{2} \)
37 \( 1 + 1.03e3T + 6.93e7T^{2} \)
41 \( 1 - 9.95e3T + 1.15e8T^{2} \)
43 \( 1 + 2.78e3T + 1.47e8T^{2} \)
47 \( 1 + 2.55e3T + 2.29e8T^{2} \)
53 \( 1 - 1.13e4T + 4.18e8T^{2} \)
59 \( 1 - 3.46e4T + 7.14e8T^{2} \)
61 \( 1 - 8.24e3T + 8.44e8T^{2} \)
67 \( 1 + 1.75e4T + 1.35e9T^{2} \)
71 \( 1 + 1.90e4T + 1.80e9T^{2} \)
73 \( 1 + 5.22e4T + 2.07e9T^{2} \)
79 \( 1 + 2.70e4T + 3.07e9T^{2} \)
83 \( 1 + 4.18e4T + 3.93e9T^{2} \)
89 \( 1 + 9.38e4T + 5.58e9T^{2} \)
97 \( 1 + 1.64e4T + 8.58e9T^{2} \)
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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

−11.23827163639329255782207383717, −10.58060806370395204539822974620, −8.676694493240489459129001180035, −7.03307955255212879736421347669, −6.16738500053545199408542287266, −5.71706734296480724850753662888, −4.32008652434463041245931443570, −3.86964170333328808833306783554, −1.86556562789027404657514160381, 0, 1.86556562789027404657514160381, 3.86964170333328808833306783554, 4.32008652434463041245931443570, 5.71706734296480724850753662888, 6.16738500053545199408542287266, 7.03307955255212879736421347669, 8.676694493240489459129001180035, 10.58060806370395204539822974620, 11.23827163639329255782207383717

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