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Properties

Label 2-245-7.4-c3-0-1
Degree $2$
Conductor $245$
Sign $0.198 - 0.980i$
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  + (−0.828 − 1.43i)2-s + (−0.166 + 0.288i)3-s + (2.62 − 4.55i)4-s + (−2.5 − 4.33i)5-s + 0.551·6-s − 21.9·8-s + (13.4 + 23.2i)9-s + (−4.14 + 7.17i)10-s + (−34.7 + 60.2i)11-s + (0.875 + 1.51i)12-s − 68.4·13-s + 1.66·15-s + (−2.85 − 4.93i)16-s + (52.1 − 90.3i)17-s + (22.2 − 38.5i)18-s + (35.9 + 62.2i)19-s + ⋯
L(s)  = 1  + (−0.292 − 0.507i)2-s + (−0.0320 + 0.0554i)3-s + (0.328 − 0.569i)4-s + (−0.223 − 0.387i)5-s + 0.0375·6-s − 0.970·8-s + (0.497 + 0.862i)9-s + (−0.130 + 0.226i)10-s + (−0.953 + 1.65i)11-s + (0.0210 + 0.0364i)12-s − 1.45·13-s + 0.0286·15-s + (−0.0445 − 0.0771i)16-s + (0.744 − 1.28i)17-s + (0.291 − 0.504i)18-s + (0.434 + 0.751i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.198 - 0.980i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.477750 + 0.390870i\)
\(L(\frac12)\) \(\approx\) \(0.477750 + 0.390870i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 \)
good2 \( 1 + (0.828 + 1.43i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (0.166 - 0.288i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (34.7 - 60.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 68.4T + 2.19e3T^{2} \)
17 \( 1 + (-52.1 + 90.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-35.9 - 62.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-50.5 - 87.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 114.T + 2.43e4T^{2} \)
31 \( 1 + (36.8 - 63.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-100. - 174. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 417.T + 6.89e4T^{2} \)
43 \( 1 - 311.T + 7.95e4T^{2} \)
47 \( 1 + (-74.8 - 129. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (135. - 235. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (259. - 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (109. + 190. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (40.3 - 69.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 91.0T + 3.57e5T^{2} \)
73 \( 1 + (-441. + 764. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (299. + 519. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 70.8T + 5.71e5T^{2} \)
89 \( 1 + (-401. - 695. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 145.T + 9.12e5T^{2} \)
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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

−11.97025751023737891031165929479, −10.73055032148975724227331538696, −9.851892588746107474740197889508, −9.515920959092606088462597290265, −7.63557832581118444293986075974, −7.26538347407309511944175296251, −5.33398424567777095260782441731, −4.80490816642384802691205172823, −2.73864052097011946902306800575, −1.61376493846494536196306927965, 0.25902356103102124154274418768, 2.71662929620870717726493637102, 3.70228928354622675091220048225, 5.51469811992971654391829659759, 6.57534458767473028996289423591, 7.49784868730982563592372399676, 8.276507906668545452264591162466, 9.338884782269028861262826595645, 10.52461874193023130695906238314, 11.45714813332261348298166621982

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