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Properties

Label 2-605-55.32-c1-0-43
Degree $2$
Conductor $605$
Sign $-0.868 + 0.495i$
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  + (1.72 − 1.72i)2-s + (0.422 − 0.422i)3-s − 3.91i·4-s + (−0.759 − 2.10i)5-s − 1.45i·6-s + (1.82 − 1.82i)7-s + (−3.29 − 3.29i)8-s + 2.64i·9-s + (−4.92 − 2.31i)10-s + (−1.65 − 1.65i)12-s + (1.98 + 1.98i)13-s − 6.28i·14-s + (−1.21 − 0.568i)15-s − 3.51·16-s + (−0.667 + 0.667i)17-s + (4.54 + 4.54i)18-s + ⋯
L(s)  = 1  + (1.21 − 1.21i)2-s + (0.244 − 0.244i)3-s − 1.95i·4-s + (−0.339 − 0.940i)5-s − 0.593i·6-s + (0.689 − 0.689i)7-s + (−1.16 − 1.16i)8-s + 0.880i·9-s + (−1.55 − 0.731i)10-s + (−0.478 − 0.478i)12-s + (0.550 + 0.550i)13-s − 1.67i·14-s + (−0.312 − 0.146i)15-s − 0.877·16-s + (−0.161 + 0.161i)17-s + (1.07 + 1.07i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.868 + 0.495i$
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: \((2,\ 605,\ (\ :1/2),\ -0.868 + 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759044 - 2.86445i\)
\(L(\frac12)\) \(\approx\) \(0.759044 - 2.86445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.759 + 2.10i)T \)
11 \( 1 \)
good2 \( 1 + (-1.72 + 1.72i)T - 2iT^{2} \)
3 \( 1 + (-0.422 + 0.422i)T - 3iT^{2} \)
7 \( 1 + (-1.82 + 1.82i)T - 7iT^{2} \)
13 \( 1 + (-1.98 - 1.98i)T + 13iT^{2} \)
17 \( 1 + (0.667 - 0.667i)T - 17iT^{2} \)
19 \( 1 + 4.14T + 19T^{2} \)
23 \( 1 + (-0.104 + 0.104i)T - 23iT^{2} \)
29 \( 1 + 6.94T + 29T^{2} \)
31 \( 1 - 9.06T + 31T^{2} \)
37 \( 1 + (-5.37 - 5.37i)T + 37iT^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 + (3.91 + 3.91i)T + 43iT^{2} \)
47 \( 1 + (0.747 + 0.747i)T + 47iT^{2} \)
53 \( 1 + (-2.91 + 2.91i)T - 53iT^{2} \)
59 \( 1 + 6.48iT - 59T^{2} \)
61 \( 1 + 9.52iT - 61T^{2} \)
67 \( 1 + (-2.94 - 2.94i)T + 67iT^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (1.25 + 1.25i)T + 73iT^{2} \)
79 \( 1 + 4.34T + 79T^{2} \)
83 \( 1 + (-5.86 - 5.86i)T + 83iT^{2} \)
89 \( 1 - 4.23iT - 89T^{2} \)
97 \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \)
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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

−10.71753474946901930039042524336, −9.728593191628926739198034327412, −8.483736149033453060690090080229, −7.82190886079173429999526362834, −6.38271446231097687468590402193, −5.10034521147927101724904773407, −4.52168757351092460737252919309, −3.73827311959819429286869652624, −2.21070531625148292390723540127, −1.27575049546199171537063522404, 2.66854917454199910447752228523, 3.69458866472730119267253235867, 4.49160627873919350652061976407, 5.75716810675018171603395833402, 6.31201855408341803078875381172, 7.26661790358761689739330379546, 8.129277300740536464622866198703, 8.866932634332945431753914683416, 10.20098704770591311670367221498, 11.30479395263296137211649089196

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