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Properties

Label 2-605-55.43-c1-0-31
Degree $2$
Conductor $605$
Sign $0.825 + 0.564i$
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  + (0.436 + 0.436i)2-s + (1.06 + 1.06i)3-s − 1.61i·4-s + (−1.07 − 1.95i)5-s + 0.928i·6-s + (1.08 + 1.08i)7-s + (1.58 − 1.58i)8-s − 0.741i·9-s + (0.384 − 1.32i)10-s + (1.71 − 1.71i)12-s + (0.147 − 0.147i)13-s + 0.950i·14-s + (0.935 − 3.22i)15-s − 1.85·16-s + (−2.69 − 2.69i)17-s + (0.324 − 0.324i)18-s + ⋯
L(s)  = 1  + (0.308 + 0.308i)2-s + (0.613 + 0.613i)3-s − 0.809i·4-s + (−0.482 − 0.876i)5-s + 0.378i·6-s + (0.411 + 0.411i)7-s + (0.558 − 0.558i)8-s − 0.247i·9-s + (0.121 − 0.419i)10-s + (0.496 − 0.496i)12-s + (0.0407 − 0.0407i)13-s + 0.254i·14-s + (0.241 − 0.833i)15-s − 0.464·16-s + (−0.653 − 0.653i)17-s + (0.0763 − 0.0763i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92990 - 0.596849i\)
\(L(\frac12)\) \(\approx\) \(1.92990 - 0.596849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.07 + 1.95i)T \)
11 \( 1 \)
good2 \( 1 + (-0.436 - 0.436i)T + 2iT^{2} \)
3 \( 1 + (-1.06 - 1.06i)T + 3iT^{2} \)
7 \( 1 + (-1.08 - 1.08i)T + 7iT^{2} \)
13 \( 1 + (-0.147 + 0.147i)T - 13iT^{2} \)
17 \( 1 + (2.69 + 2.69i)T + 17iT^{2} \)
19 \( 1 - 6.52T + 19T^{2} \)
23 \( 1 + (4.95 + 4.95i)T + 23iT^{2} \)
29 \( 1 - 9.03T + 29T^{2} \)
31 \( 1 - 6.06T + 31T^{2} \)
37 \( 1 + (0.916 - 0.916i)T - 37iT^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 + (7.33 - 7.33i)T - 43iT^{2} \)
47 \( 1 + (-0.236 + 0.236i)T - 47iT^{2} \)
53 \( 1 + (-3.74 - 3.74i)T + 53iT^{2} \)
59 \( 1 - 5.21iT - 59T^{2} \)
61 \( 1 - 14.7iT - 61T^{2} \)
67 \( 1 + (1.11 - 1.11i)T - 67iT^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + (9.27 - 9.27i)T - 73iT^{2} \)
79 \( 1 + 5.52T + 79T^{2} \)
83 \( 1 + (-5.84 + 5.84i)T - 83iT^{2} \)
89 \( 1 - 9.01iT - 89T^{2} \)
97 \( 1 + (-5.99 + 5.99i)T - 97iT^{2} \)
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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

−10.25020124176996901440386652871, −9.707874357254687494597877979019, −8.791693603076535721477940282675, −8.189251888072418399950698521606, −6.91156238612882723137626007599, −5.85205459836306155228584027188, −4.76700465831599528370259226371, −4.33455868452495984804783555028, −2.83516789692715030586443552158, −1.04974089043472879081594953342, 1.89887213093974475708846204701, 2.97955151863817336192718728870, 3.80930992764254501830118606739, 4.92640810663203033497721456262, 6.55161836056613283901884280130, 7.43534401895919281876764697375, 7.926875864590382829980339315042, 8.619975104624273393981996759254, 10.07901268359507942320547375251, 10.89458957196017441338986640166

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