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Properties

Label 2-605-55.28-c1-0-28
Degree $2$
Conductor $605$
Sign $0.932 + 0.361i$
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  + (−0.690 + 1.35i)2-s + (−0.124 − 0.787i)3-s + (−0.182 − 0.251i)4-s + (0.543 − 2.16i)5-s + (1.15 + 0.374i)6-s + (0.189 + 0.0299i)7-s + (−2.53 + 0.401i)8-s + (2.24 − 0.730i)9-s + (2.56 + 2.23i)10-s + (−0.175 + 0.175i)12-s + (−2.75 − 1.40i)13-s + (−0.171 + 0.235i)14-s + (−1.77 − 0.157i)15-s + (1.39 − 4.30i)16-s + (3.30 − 1.68i)17-s + (−0.562 + 3.54i)18-s + ⋯
L(s)  = 1  + (−0.487 + 0.957i)2-s + (−0.0720 − 0.454i)3-s + (−0.0912 − 0.125i)4-s + (0.243 − 0.969i)5-s + (0.470 + 0.152i)6-s + (0.0714 + 0.0113i)7-s + (−0.896 + 0.142i)8-s + (0.749 − 0.243i)9-s + (0.810 + 0.706i)10-s + (−0.0505 + 0.0505i)12-s + (−0.764 − 0.389i)13-s + (−0.0457 + 0.0629i)14-s + (−0.458 − 0.0407i)15-s + (0.349 − 1.07i)16-s + (0.800 − 0.408i)17-s + (−0.132 + 0.836i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.932 + 0.361i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10812 - 0.207266i\)
\(L(\frac12)\) \(\approx\) \(1.10812 - 0.207266i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.543 + 2.16i)T \)
11 \( 1 \)
good2 \( 1 + (0.690 - 1.35i)T + (-1.17 - 1.61i)T^{2} \)
3 \( 1 + (0.124 + 0.787i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.189 - 0.0299i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (2.75 + 1.40i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-3.30 + 1.68i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.601 + 0.437i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.14 + 1.14i)T + 23iT^{2} \)
29 \( 1 + (-7.72 + 5.61i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.108 - 0.333i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.845 + 5.33i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-3.93 + 5.41i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.72 - 3.72i)T - 43iT^{2} \)
47 \( 1 + (-12.2 + 1.93i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (4.09 - 8.04i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.65 - 7.78i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.60 - 1.82i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-4.13 + 4.13i)T - 67iT^{2} \)
71 \( 1 + (3.54 - 10.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.375 - 2.37i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (0.207 + 0.637i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (7.57 + 14.8i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 7.92iT - 89T^{2} \)
97 \( 1 + (-1.22 - 0.626i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.21036879908496305685310217563, −9.546511003462715562564761948179, −8.652220180936360714378804623880, −7.82655529129228819427355992611, −7.22941981965513706916976647403, −6.20911374496514872841429367238, −5.37676804197511257033762558083, −4.22310769126475450436997057399, −2.50777723733814361364234054528, −0.78163027804832959617277321979, 1.56817205717664118065300105035, 2.72176505917084268460429797331, 3.74098722706716916086634694157, 5.05366025306034286792845827848, 6.28067441328887805844269646831, 7.11056113072142495985737282204, 8.241374819605014730455461654805, 9.524519417589713585220894206623, 9.984480859208531358538062107487, 10.54525055700112592990177720807

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