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Properties

Label 2-245-5.4-c3-0-54
Degree $2$
Conductor $245$
Sign $0.814 - 0.579i$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.30i·2-s − 8.20i·3-s − 20.0·4-s + (9.10 − 6.48i)5-s − 43.4·6-s + 64.1i·8-s − 40.2·9-s + (−34.3 − 48.2i)10-s + 3.09·11-s + 164. i·12-s + 26.1i·13-s + (−53.1 − 74.7i)15-s + 179.·16-s + 73.0i·17-s + 213. i·18-s − 82.8·19-s + ⋯
L(s)  = 1  − 1.87i·2-s − 1.57i·3-s − 2.51·4-s + (0.814 − 0.579i)5-s − 2.95·6-s + 2.83i·8-s − 1.49·9-s + (−1.08 − 1.52i)10-s + 0.0848·11-s + 3.96i·12-s + 0.558i·13-s + (−0.915 − 1.28i)15-s + 2.79·16-s + 1.04i·17-s + 2.79i·18-s − 1.00·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.814 - 0.579i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.814 - 0.579i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.978264 + 0.312667i\)
\(L(\frac12)\) \(\approx\) \(0.978264 + 0.312667i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-9.10 + 6.48i)T \)
7 \( 1 \)
good2 \( 1 + 5.30iT - 8T^{2} \)
3 \( 1 + 8.20iT - 27T^{2} \)
11 \( 1 - 3.09T + 1.33e3T^{2} \)
13 \( 1 - 26.1iT - 2.19e3T^{2} \)
17 \( 1 - 73.0iT - 4.91e3T^{2} \)
19 \( 1 + 82.8T + 6.85e3T^{2} \)
23 \( 1 + 85.8iT - 1.21e4T^{2} \)
29 \( 1 + 209.T + 2.43e4T^{2} \)
31 \( 1 - 8.24T + 2.97e4T^{2} \)
37 \( 1 + 295. iT - 5.06e4T^{2} \)
41 \( 1 + 356.T + 6.89e4T^{2} \)
43 \( 1 + 148. iT - 7.95e4T^{2} \)
47 \( 1 + 444. iT - 1.03e5T^{2} \)
53 \( 1 + 323. iT - 1.48e5T^{2} \)
59 \( 1 + 422.T + 2.05e5T^{2} \)
61 \( 1 - 778.T + 2.26e5T^{2} \)
67 \( 1 + 203. iT - 3.00e5T^{2} \)
71 \( 1 - 731.T + 3.57e5T^{2} \)
73 \( 1 - 586. iT - 3.89e5T^{2} \)
79 \( 1 + 208.T + 4.93e5T^{2} \)
83 \( 1 - 396. iT - 5.71e5T^{2} \)
89 \( 1 + 571.T + 7.04e5T^{2} \)
97 \( 1 - 846. iT - 9.12e5T^{2} \)
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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

−10.99551269617738849294888134537, −10.03284770668976802413124695644, −8.893477947349105408210951918061, −8.297445610684865262602361965054, −6.64363302229633220006393115443, −5.42237208289031119820569707802, −3.94815682517920084714356274479, −2.18216831635639570384550288047, −1.75545822918277429558533571940, −0.40863336843961682670004336449, 3.35494891216217378739899923484, 4.64924542488058029407581288600, 5.43439578776616710541641000453, 6.31401541496922214102050164700, 7.46160796358881375335248501587, 8.692089975916618825721815356607, 9.516778064151913002475593065272, 10.05880722781296057452666348515, 11.17951454614336716667252639985, 13.05827196004771728714276917476

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