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Properties

Label 2-605-55.49-c1-0-9
Degree $2$
Conductor $605$
Sign $-0.451 - 0.892i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (−1.92 + 0.625i)2-s + (−1.54 + 2.12i)3-s + (1.69 − 1.23i)4-s + (−2.19 − 0.404i)5-s + (1.63 − 5.04i)6-s + (0.567 + 0.781i)7-s + (−0.113 + 0.156i)8-s + (−1.19 − 3.67i)9-s + (4.48 − 0.596i)10-s + 5.49i·12-s + (4.30 − 1.39i)13-s + (−1.58 − 1.14i)14-s + (4.24 − 4.03i)15-s + (−1.17 + 3.61i)16-s + (3.17 + 1.03i)17-s + (4.60 + 6.33i)18-s + ⋯
L(s)  = 1  + (−1.36 + 0.442i)2-s + (−0.889 + 1.22i)3-s + (0.847 − 0.615i)4-s + (−0.983 − 0.180i)5-s + (0.669 − 2.05i)6-s + (0.214 + 0.295i)7-s + (−0.0400 + 0.0551i)8-s + (−0.398 − 1.22i)9-s + (1.41 − 0.188i)10-s + 1.58i·12-s + (1.19 − 0.387i)13-s + (−0.422 − 0.307i)14-s + (1.09 − 1.04i)15-s + (−0.293 + 0.903i)16-s + (0.769 + 0.250i)17-s + (1.08 + 1.49i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228576 + 0.371834i\)
\(L(\frac12)\) \(\approx\) \(0.228576 + 0.371834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 + 0.404i)T \)
11 \( 1 \)
good2 \( 1 + (1.92 - 0.625i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.54 - 2.12i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.567 - 0.781i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.30 + 1.39i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.17 - 1.03i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.65 - 1.92i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.36iT - 23T^{2} \)
29 \( 1 + (-3.97 + 2.88i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.129 + 0.397i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.72 - 5.12i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.68 + 3.40i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.26iT - 43T^{2} \)
47 \( 1 + (2.54 - 3.49i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.53 + 0.822i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.19 - 5.95i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.763 - 2.34i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.60iT - 67T^{2} \)
71 \( 1 + (1.68 - 5.18i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.843 + 1.16i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.310 + 0.954i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-7.03 - 2.28i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (2.87 - 0.932i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.54160533174295488094428288797, −10.21256806672281734957537592142, −9.145988963401470162232074364039, −8.390890617543084258943253327904, −7.75290237133103999158214034979, −6.49430562388179542989703606143, −5.56152353366485705984242009204, −4.43568988327687423667221626849, −3.52049251379731163085012320514, −0.914514931985444580197476536087, 0.66522148712542128584505738348, 1.56358083884272239683303847496, 3.27366760871032794950759810295, 4.89429889779732773861447590648, 6.20323882512504012958940876707, 7.19667140065077008145785743673, 7.69178582560196440678118753040, 8.463135744371175602007574613486, 9.469273529414378914536461985462, 10.67700556356960851003878043528

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