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Properties

Label 2-605-55.32-c1-0-28
Degree $2$
Conductor $605$
Sign $0.794 + 0.606i$
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  + (−0.503 + 0.503i)2-s + (−1.98 + 1.98i)3-s + 1.49i·4-s + (−1.36 + 1.77i)5-s − 2.00i·6-s + (2.43 − 2.43i)7-s + (−1.75 − 1.75i)8-s − 4.91i·9-s + (−0.204 − 1.57i)10-s + (−2.97 − 2.97i)12-s + (−2.88 − 2.88i)13-s + 2.44i·14-s + (−0.809 − 6.23i)15-s − 1.21·16-s + (0.334 − 0.334i)17-s + (2.47 + 2.47i)18-s + ⋯
L(s)  = 1  + (−0.355 + 0.355i)2-s + (−1.14 + 1.14i)3-s + 0.746i·4-s + (−0.610 + 0.792i)5-s − 0.817i·6-s + (0.919 − 0.919i)7-s + (−0.621 − 0.621i)8-s − 1.63i·9-s + (−0.0647 − 0.498i)10-s + (−0.857 − 0.857i)12-s + (−0.801 − 0.801i)13-s + 0.654i·14-s + (−0.208 − 1.61i)15-s − 0.304·16-s + (0.0812 − 0.0812i)17-s + (0.582 + 0.582i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.794 + 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171074 - 0.0578598i\)
\(L(\frac12)\) \(\approx\) \(0.171074 - 0.0578598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.36 - 1.77i)T \)
11 \( 1 \)
good2 \( 1 + (0.503 - 0.503i)T - 2iT^{2} \)
3 \( 1 + (1.98 - 1.98i)T - 3iT^{2} \)
7 \( 1 + (-2.43 + 2.43i)T - 7iT^{2} \)
13 \( 1 + (2.88 + 2.88i)T + 13iT^{2} \)
17 \( 1 + (-0.334 + 0.334i)T - 17iT^{2} \)
19 \( 1 + 4.06T + 19T^{2} \)
23 \( 1 + (-0.771 + 0.771i)T - 23iT^{2} \)
29 \( 1 - 2.39T + 29T^{2} \)
31 \( 1 - 3.82T + 31T^{2} \)
37 \( 1 + (3.60 + 3.60i)T + 37iT^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + (7.82 + 7.82i)T + 43iT^{2} \)
47 \( 1 + (-2.80 - 2.80i)T + 47iT^{2} \)
53 \( 1 + (3.61 - 3.61i)T - 53iT^{2} \)
59 \( 1 - 7.63iT - 59T^{2} \)
61 \( 1 + 3.34iT - 61T^{2} \)
67 \( 1 + (1.52 + 1.52i)T + 67iT^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + (3.02 + 3.02i)T + 73iT^{2} \)
79 \( 1 + 6.48T + 79T^{2} \)
83 \( 1 + (7.68 + 7.68i)T + 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (0.0994 + 0.0994i)T + 97iT^{2} \)
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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

−10.42818123798743442721848754631, −10.13444844293733198616841820748, −8.720020612891066706699573835402, −7.78715827929332978918450818322, −7.10016752971448315079761191686, −6.11149733281998811228927209261, −4.73890125679674580714258591168, −4.18879320550380719255337462442, −3.06790939770410056991109561081, −0.13874880303027450907919224292, 1.32880545748211048829935582024, 2.17858144971781687439766553979, 4.69227964661671291356396276174, 5.23413122044654948063161974001, 6.16818753571235725035635877072, 7.09594665012790806876947578033, 8.243173993874289204233291140157, 8.834055259067138559195567136132, 10.00845564794433642513610474673, 11.11548058267789928435688504725

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