Properties

Label 2-245-1.1-c3-0-40
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 2·3-s + 4-s − 5·5-s + 6·6-s − 21·8-s − 23·9-s − 15·10-s − 45·11-s + 2·12-s − 59·13-s − 10·15-s − 71·16-s + 54·17-s − 69·18-s + 121·19-s − 5·20-s − 135·22-s + 69·23-s − 42·24-s + 25·25-s − 177·26-s − 100·27-s − 162·29-s − 30·30-s + 88·31-s − 45·32-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.384·3-s + 1/8·4-s − 0.447·5-s + 0.408·6-s − 0.928·8-s − 0.851·9-s − 0.474·10-s − 1.23·11-s + 0.0481·12-s − 1.25·13-s − 0.172·15-s − 1.10·16-s + 0.770·17-s − 0.903·18-s + 1.46·19-s − 0.0559·20-s − 1.30·22-s + 0.625·23-s − 0.357·24-s + 1/5·25-s − 1.33·26-s − 0.712·27-s − 1.03·29-s − 0.182·30-s + 0.509·31-s − 0.248·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p T \)
7 \( 1 \)
good2 \( 1 - 3 T + p^{3} T^{2} \)
3 \( 1 - 2 T + p^{3} T^{2} \)
11 \( 1 + 45 T + p^{3} T^{2} \)
13 \( 1 + 59 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 - 121 T + p^{3} T^{2} \)
23 \( 1 - 3 p T + p^{3} T^{2} \)
29 \( 1 + 162 T + p^{3} T^{2} \)
31 \( 1 - 88 T + p^{3} T^{2} \)
37 \( 1 + 7 p T + p^{3} T^{2} \)
41 \( 1 + 195 T + p^{3} T^{2} \)
43 \( 1 + 286 T + p^{3} T^{2} \)
47 \( 1 + 45 T + p^{3} T^{2} \)
53 \( 1 - 597 T + p^{3} T^{2} \)
59 \( 1 - 360 T + p^{3} T^{2} \)
61 \( 1 + 392 T + p^{3} T^{2} \)
67 \( 1 + 280 T + p^{3} T^{2} \)
71 \( 1 - 48 T + p^{3} T^{2} \)
73 \( 1 + 668 T + p^{3} T^{2} \)
79 \( 1 - 782 T + p^{3} T^{2} \)
83 \( 1 + 768 T + p^{3} T^{2} \)
89 \( 1 - 1194 T + p^{3} T^{2} \)
97 \( 1 + 902 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.63017361638978740889091580292, −10.23339313000226430173744263134, −9.199230036378017995700260436697, −8.098714730841790071468302911584, −7.18113473606970315864530087313, −5.48961337171956711852924490803, −5.04826302964011918732113428635, −3.47421677591272543079165638432, −2.70471734633181131211946638333, 0, 2.70471734633181131211946638333, 3.47421677591272543079165638432, 5.04826302964011918732113428635, 5.48961337171956711852924490803, 7.18113473606970315864530087313, 8.098714730841790071468302911584, 9.199230036378017995700260436697, 10.23339313000226430173744263134, 11.63017361638978740889091580292

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