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Properties

Label 2-245-245.223-c1-0-17
Degree $2$
Conductor $245$
Sign $0.970 + 0.239i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.462 − 1.32i)2-s + (1.63 + 2.59i)3-s + (0.0310 + 0.0247i)4-s + (−0.694 − 2.12i)5-s + (4.18 − 0.955i)6-s + (1.58 − 2.11i)7-s + (2.41 − 1.51i)8-s + (−2.78 + 5.77i)9-s + (−3.13 − 0.0648i)10-s + (−4.36 + 2.10i)11-s + (−0.0136 + 0.121i)12-s + (−0.493 + 1.41i)13-s + (−2.06 − 3.07i)14-s + (4.38 − 5.27i)15-s + (−0.872 − 3.82i)16-s + (−0.0228 + 0.202i)17-s + ⋯
L(s)  = 1  + (0.326 − 0.934i)2-s + (0.942 + 1.49i)3-s + (0.0155 + 0.0123i)4-s + (−0.310 − 0.950i)5-s + (1.70 − 0.390i)6-s + (0.599 − 0.800i)7-s + (0.854 − 0.537i)8-s + (−0.927 + 1.92i)9-s + (−0.989 − 0.0205i)10-s + (−1.31 + 0.633i)11-s + (−0.00394 + 0.0349i)12-s + (−0.136 + 0.391i)13-s + (−0.551 − 0.822i)14-s + (1.13 − 1.36i)15-s + (−0.218 − 0.955i)16-s + (−0.00554 + 0.0492i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.970 + 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97679 - 0.240528i\)
\(L(\frac12)\) \(\approx\) \(1.97679 - 0.240528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.694 + 2.12i)T \)
7 \( 1 + (-1.58 + 2.11i)T \)
good2 \( 1 + (-0.462 + 1.32i)T + (-1.56 - 1.24i)T^{2} \)
3 \( 1 + (-1.63 - 2.59i)T + (-1.30 + 2.70i)T^{2} \)
11 \( 1 + (4.36 - 2.10i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.493 - 1.41i)T + (-10.1 - 8.10i)T^{2} \)
17 \( 1 + (0.0228 - 0.202i)T + (-16.5 - 3.78i)T^{2} \)
19 \( 1 - 3.88T + 19T^{2} \)
23 \( 1 + (6.65 - 0.749i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (0.441 - 0.351i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 + 8.44iT - 31T^{2} \)
37 \( 1 + (-6.29 - 0.709i)T + (36.0 + 8.23i)T^{2} \)
41 \( 1 + (5.11 + 1.16i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (5.85 - 9.31i)T + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (5.30 + 1.85i)T + (36.7 + 29.3i)T^{2} \)
53 \( 1 + (-4.21 + 0.474i)T + (51.6 - 11.7i)T^{2} \)
59 \( 1 + (0.902 + 3.95i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-2.34 + 1.87i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (2.29 - 2.29i)T - 67iT^{2} \)
71 \( 1 + (-0.892 + 1.11i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-9.49 + 3.32i)T + (57.0 - 45.5i)T^{2} \)
79 \( 1 - 5.77iT - 79T^{2} \)
83 \( 1 + (-1.88 + 0.661i)T + (64.8 - 51.7i)T^{2} \)
89 \( 1 + (-7.34 - 3.53i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (6.10 + 6.10i)T + 97iT^{2} \)
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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

−11.82745677222437158323035067392, −11.07231407768823840566298597903, −10.00102911272837390945233526995, −9.680259595635158686587450388801, −8.104314657281611722983855089380, −7.71348647070710718081991019038, −5.04808152389838735871810644544, −4.41090254298693281913003163692, −3.56554893422153970242151583140, −2.13838675686279483204184252845, 2.07378614724320140125966611798, 3.07927579723492121419511692256, 5.36699165535356292672268102203, 6.28084149488083004452452896889, 7.28496740201205073183962198746, 7.938126964925709221547092973206, 8.467423865540540189612587747734, 10.27200620127021641034899803286, 11.42643660032811731909161609887, 12.27987706400129637728920254855

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