Properties

Label 2-245-1.1-c3-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.58·2-s − 6.65·3-s − 1.31·4-s + 5·5-s − 17.2·6-s − 24.0·8-s + 17.3·9-s + 12.9·10-s + 38.2·11-s + 8.74·12-s − 19.3·13-s − 33.2·15-s − 51.7·16-s + 87.2·17-s + 44.7·18-s + 44.2·19-s − 6.56·20-s + 98.9·22-s + 218.·23-s + 160.·24-s + 25·25-s − 50.0·26-s + 64.4·27-s − 46.9·29-s − 86.0·30-s − 194.·31-s + 58.8·32-s + ⋯
L(s)  = 1  + 0.914·2-s − 1.28·3-s − 0.164·4-s + 0.447·5-s − 1.17·6-s − 1.06·8-s + 0.641·9-s + 0.408·10-s + 1.04·11-s + 0.210·12-s − 0.412·13-s − 0.572·15-s − 0.808·16-s + 1.24·17-s + 0.586·18-s + 0.534·19-s − 0.0734·20-s + 0.958·22-s + 1.97·23-s + 1.36·24-s + 0.200·25-s − 0.377·26-s + 0.459·27-s − 0.300·29-s − 0.523·30-s − 1.12·31-s + 0.324·32-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: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.712953511\)
\(L(\frac12)\) \(\approx\) \(1.712953511\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.58T + 8T^{2} \)
3 \( 1 + 6.65T + 27T^{2} \)
11 \( 1 - 38.2T + 1.33e3T^{2} \)
13 \( 1 + 19.3T + 2.19e3T^{2} \)
17 \( 1 - 87.2T + 4.91e3T^{2} \)
19 \( 1 - 44.2T + 6.85e3T^{2} \)
23 \( 1 - 218.T + 1.21e4T^{2} \)
29 \( 1 + 46.9T + 2.43e4T^{2} \)
31 \( 1 + 194.T + 2.97e4T^{2} \)
37 \( 1 - 366.T + 5.06e4T^{2} \)
41 \( 1 - 339.T + 6.89e4T^{2} \)
43 \( 1 + 226.T + 7.95e4T^{2} \)
47 \( 1 + 11.6T + 1.03e5T^{2} \)
53 \( 1 + 209.T + 1.48e5T^{2} \)
59 \( 1 - 616T + 2.05e5T^{2} \)
61 \( 1 + 320.T + 2.26e5T^{2} \)
67 \( 1 - 14.5T + 3.00e5T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 + 824.T + 3.89e5T^{2} \)
79 \( 1 - 156.T + 4.93e5T^{2} \)
83 \( 1 - 1.03e3T + 5.71e5T^{2} \)
89 \( 1 - 170.T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−11.78337554304876739216182815724, −11.07299026007667274256709614166, −9.775216624722443456069907555346, −9.000028564575324861953296554889, −7.24239593767487482425960679859, −6.12689537594100828896040468733, −5.45678844671800828457157010558, −4.58657810812247511058662225949, −3.19278770035578903475279401123, −0.928496654100638927623280253138, 0.928496654100638927623280253138, 3.19278770035578903475279401123, 4.58657810812247511058662225949, 5.45678844671800828457157010558, 6.12689537594100828896040468733, 7.24239593767487482425960679859, 9.000028564575324861953296554889, 9.775216624722443456069907555346, 11.07299026007667274256709614166, 11.78337554304876739216182815724

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