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Properties

Label 2-245-1.1-c1-0-4
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s + 5-s + 6·6-s + 6·9-s − 2·10-s + 11-s − 6·12-s − 3·13-s − 3·15-s − 4·16-s + 3·17-s − 12·18-s − 6·19-s + 2·20-s − 2·22-s − 4·23-s + 25-s + 6·26-s − 9·27-s − 29-s + 6·30-s − 6·31-s + 8·32-s − 3·33-s − 6·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s + 0.447·5-s + 2.44·6-s + 2·9-s − 0.632·10-s + 0.301·11-s − 1.73·12-s − 0.832·13-s − 0.774·15-s − 16-s + 0.727·17-s − 2.82·18-s − 1.37·19-s + 0.447·20-s − 0.426·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s − 1.73·27-s − 0.185·29-s + 1.09·30-s − 1.07·31-s + 1.41·32-s − 0.522·33-s − 1.02·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−11.36832909695861645450710523451, −10.39298032670518976720614402649, −10.02792671861300266401979001834, −8.895041085769743028052283031023, −7.56848203557801886905576002740, −6.66812515626338690414111791742, −5.69074574951383439724683067976, −4.49258853804675338518966926948, −1.70898778880961935771596537039, 0, 1.70898778880961935771596537039, 4.49258853804675338518966926948, 5.69074574951383439724683067976, 6.66812515626338690414111791742, 7.56848203557801886905576002740, 8.895041085769743028052283031023, 10.02792671861300266401979001834, 10.39298032670518976720614402649, 11.36832909695861645450710523451

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