Properties

Label 2-245-1.1-c3-0-13
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  − 2.31·2-s + 5·3-s − 2.63·4-s + 5·5-s − 11.5·6-s + 24.6·8-s − 2·9-s − 11.5·10-s + 46.2·11-s − 13.1·12-s − 61.3·13-s + 25·15-s − 36·16-s + 101.·17-s + 4.63·18-s + 3.66·19-s − 13.1·20-s − 107.·22-s + 84.8·23-s + 123.·24-s + 25·25-s + 142.·26-s − 145·27-s + 30.1·29-s − 57.9·30-s + 188.·31-s − 113.·32-s + ⋯
L(s)  = 1  − 0.819·2-s + 0.962·3-s − 0.329·4-s + 0.447·5-s − 0.788·6-s + 1.08·8-s − 0.0740·9-s − 0.366·10-s + 1.26·11-s − 0.316·12-s − 1.30·13-s + 0.430·15-s − 0.562·16-s + 1.44·17-s + 0.0606·18-s + 0.0442·19-s − 0.147·20-s − 1.03·22-s + 0.769·23-s + 1.04·24-s + 0.200·25-s + 1.07·26-s − 1.03·27-s + 0.193·29-s − 0.352·30-s + 1.09·31-s − 0.627·32-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

\(L(2)\) \(\approx\) \(1.594236251\)
\(L(\frac12)\) \(\approx\) \(1.594236251\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.31T + 8T^{2} \)
3 \( 1 - 5T + 27T^{2} \)
11 \( 1 - 46.2T + 1.33e3T^{2} \)
13 \( 1 + 61.3T + 2.19e3T^{2} \)
17 \( 1 - 101.T + 4.91e3T^{2} \)
19 \( 1 - 3.66T + 6.85e3T^{2} \)
23 \( 1 - 84.8T + 1.21e4T^{2} \)
29 \( 1 - 30.1T + 2.43e4T^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 - 18.0T + 5.06e4T^{2} \)
41 \( 1 - 481.T + 6.89e4T^{2} \)
43 \( 1 + 97.7T + 7.95e4T^{2} \)
47 \( 1 - 117.T + 1.03e5T^{2} \)
53 \( 1 - 667.T + 1.48e5T^{2} \)
59 \( 1 - 57.3T + 2.05e5T^{2} \)
61 \( 1 - 738.T + 2.26e5T^{2} \)
67 \( 1 - 552.T + 3.00e5T^{2} \)
71 \( 1 + 740.T + 3.57e5T^{2} \)
73 \( 1 - 233.T + 3.89e5T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 - 683.T + 5.71e5T^{2} \)
89 \( 1 + 1.38e3T + 7.04e5T^{2} \)
97 \( 1 - 218.T + 9.12e5T^{2} \)
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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

−11.58968204306984027042945458344, −10.15671253723971945814386540201, −9.561655027061106917618279824170, −8.878326135696422137222674198250, −7.967570444818549368119420447570, −7.03911760226393773621923369053, −5.44027596967170726333397173976, −4.06453919960310431312638423250, −2.62346085043667259630895313482, −1.07303701354296771552814746054, 1.07303701354296771552814746054, 2.62346085043667259630895313482, 4.06453919960310431312638423250, 5.44027596967170726333397173976, 7.03911760226393773621923369053, 7.967570444818549368119420447570, 8.878326135696422137222674198250, 9.561655027061106917618279824170, 10.15671253723971945814386540201, 11.58968204306984027042945458344

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