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Properties

Label 2-605-55.32-c1-0-25
Degree $2$
Conductor $605$
Sign $0.839 - 0.543i$
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  + (0.125 − 0.125i)2-s + (1.65 − 1.65i)3-s + 1.96i·4-s + (2.04 + 0.909i)5-s − 0.416i·6-s + (−3.15 + 3.15i)7-s + (0.498 + 0.498i)8-s − 2.49i·9-s + (0.371 − 0.142i)10-s + (3.26 + 3.26i)12-s + (1.97 + 1.97i)13-s + 0.793i·14-s + (4.89 − 1.87i)15-s − 3.81·16-s + (2.20 − 2.20i)17-s + (−0.313 − 0.313i)18-s + ⋯
L(s)  = 1  + (0.0888 − 0.0888i)2-s + (0.957 − 0.957i)3-s + 0.984i·4-s + (0.913 + 0.406i)5-s − 0.170i·6-s + (−1.19 + 1.19i)7-s + (0.176 + 0.176i)8-s − 0.832i·9-s + (0.117 − 0.0450i)10-s + (0.941 + 0.941i)12-s + (0.546 + 0.546i)13-s + 0.212i·14-s + (1.26 − 0.484i)15-s − 0.952·16-s + (0.534 − 0.534i)17-s + (−0.0739 − 0.0739i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.839 - 0.543i$
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: \((2,\ 605,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09326 + 0.619078i\)
\(L(\frac12)\) \(\approx\) \(2.09326 + 0.619078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.04 - 0.909i)T \)
11 \( 1 \)
good2 \( 1 + (-0.125 + 0.125i)T - 2iT^{2} \)
3 \( 1 + (-1.65 + 1.65i)T - 3iT^{2} \)
7 \( 1 + (3.15 - 3.15i)T - 7iT^{2} \)
13 \( 1 + (-1.97 - 1.97i)T + 13iT^{2} \)
17 \( 1 + (-2.20 + 2.20i)T - 17iT^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
23 \( 1 + (-1.27 + 1.27i)T - 23iT^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + 5.13T + 31T^{2} \)
37 \( 1 + (1.75 + 1.75i)T + 37iT^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 + (4.31 + 4.31i)T + 43iT^{2} \)
47 \( 1 + (-2.65 - 2.65i)T + 47iT^{2} \)
53 \( 1 + (-0.315 + 0.315i)T - 53iT^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 1.38iT - 61T^{2} \)
67 \( 1 + (-0.721 - 0.721i)T + 67iT^{2} \)
71 \( 1 - 3.52T + 71T^{2} \)
73 \( 1 + (-6.58 - 6.58i)T + 73iT^{2} \)
79 \( 1 - 6.02T + 79T^{2} \)
83 \( 1 + (3.45 + 3.45i)T + 83iT^{2} \)
89 \( 1 - 2.57iT - 89T^{2} \)
97 \( 1 + (4.62 + 4.62i)T + 97iT^{2} \)
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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

−10.72701282767798955945841960436, −9.464718229302589587517600623345, −8.938547970427228718726380285780, −8.240383799288780063807036497541, −7.00348943513188821069819817962, −6.59927919627715503426642262134, −5.40765381972864746521493824085, −3.54531143287217265556073099794, −2.75475088238670104864867977528, −2.06699514440091358641131429799, 1.15899467005400552062886140072, 2.90458555996995175442367371647, 3.91134894406186766274664023833, 4.90555030872754044862135294582, 6.04374367809864813160451227340, 6.74620209523721718942846237802, 8.197478127258659894303785007242, 9.167102517616851519606568137766, 9.814932559200773612077008500066, 10.24676340504755553494812070644

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