Properties

Label 2-245-1.1-c3-0-20
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.51·2-s − 3.86·3-s + 22.4·4-s + 5·5-s + 21.3·6-s − 79.7·8-s − 12.0·9-s − 27.5·10-s + 34.5·11-s − 86.6·12-s − 68.8·13-s − 19.3·15-s + 260.·16-s + 91.4·17-s + 66.7·18-s − 11.8·19-s + 112.·20-s − 190.·22-s − 0.104·23-s + 307.·24-s + 25·25-s + 380.·26-s + 150.·27-s + 190.·29-s + 106.·30-s − 159.·31-s − 798.·32-s + ⋯
L(s)  = 1  − 1.95·2-s − 0.742·3-s + 2.80·4-s + 0.447·5-s + 1.44·6-s − 3.52·8-s − 0.448·9-s − 0.872·10-s + 0.946·11-s − 2.08·12-s − 1.46·13-s − 0.332·15-s + 4.06·16-s + 1.30·17-s + 0.874·18-s − 0.142·19-s + 1.25·20-s − 1.84·22-s − 0.000944·23-s + 2.61·24-s + 0.200·25-s + 2.86·26-s + 1.07·27-s + 1.22·29-s + 0.648·30-s − 0.925·31-s − 4.41·32-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: no
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 \( 1 - 5T \)
7 \( 1 \)
good2 \( 1 + 5.51T + 8T^{2} \)
3 \( 1 + 3.86T + 27T^{2} \)
11 \( 1 - 34.5T + 1.33e3T^{2} \)
13 \( 1 + 68.8T + 2.19e3T^{2} \)
17 \( 1 - 91.4T + 4.91e3T^{2} \)
19 \( 1 + 11.8T + 6.85e3T^{2} \)
23 \( 1 + 0.104T + 1.21e4T^{2} \)
29 \( 1 - 190.T + 2.43e4T^{2} \)
31 \( 1 + 159.T + 2.97e4T^{2} \)
37 \( 1 + 177.T + 5.06e4T^{2} \)
41 \( 1 + 145.T + 6.89e4T^{2} \)
43 \( 1 - 8.25T + 7.95e4T^{2} \)
47 \( 1 + 260.T + 1.03e5T^{2} \)
53 \( 1 - 353.T + 1.48e5T^{2} \)
59 \( 1 + 240.T + 2.05e5T^{2} \)
61 \( 1 + 778.T + 2.26e5T^{2} \)
67 \( 1 - 151.T + 3.00e5T^{2} \)
71 \( 1 + 311.T + 3.57e5T^{2} \)
73 \( 1 + 639.T + 3.89e5T^{2} \)
79 \( 1 - 391.T + 4.93e5T^{2} \)
83 \( 1 + 493.T + 5.71e5T^{2} \)
89 \( 1 + 473.T + 7.04e5T^{2} \)
97 \( 1 + 839.T + 9.12e5T^{2} \)
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

−10.84969434584608584622602808874, −10.07993585920917841798757297326, −9.362610956585186056696318218730, −8.395217796186027266267573732729, −7.28223690739680053735564493804, −6.44313894457437787163533436949, −5.40943310041139647362628436872, −2.87480817841849403652512345829, −1.40143635319238523844250757985, 0, 1.40143635319238523844250757985, 2.87480817841849403652512345829, 5.40943310041139647362628436872, 6.44313894457437787163533436949, 7.28223690739680053735564493804, 8.395217796186027266267573732729, 9.362610956585186056696318218730, 10.07993585920917841798757297326, 10.84969434584608584622602808874

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