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Properties

Label 2-605-11.9-c1-0-16
Degree $2$
Conductor $605$
Sign $0.522 - 0.852i$
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  + (−0.148 + 0.457i)2-s + (1.29 − 0.941i)3-s + (1.43 + 1.03i)4-s + (0.309 + 0.951i)5-s + (0.238 + 0.732i)6-s + (−0.389 − 0.282i)7-s + (−1.46 + 1.06i)8-s + (−0.134 + 0.413i)9-s − 0.480·10-s + 2.83·12-s + (−1.48 + 4.56i)13-s + (0.187 − 0.135i)14-s + (1.29 + 0.941i)15-s + (0.823 + 2.53i)16-s + (0.773 + 2.37i)17-s + (−0.169 − 0.122i)18-s + ⋯
L(s)  = 1  + (−0.105 + 0.323i)2-s + (0.748 − 0.543i)3-s + (0.715 + 0.519i)4-s + (0.138 + 0.425i)5-s + (0.0971 + 0.299i)6-s + (−0.147 − 0.106i)7-s + (−0.518 + 0.376i)8-s + (−0.0447 + 0.137i)9-s − 0.152·10-s + 0.817·12-s + (−0.411 + 1.26i)13-s + (0.0500 − 0.0363i)14-s + (0.334 + 0.243i)15-s + (0.205 + 0.633i)16-s + (0.187 + 0.577i)17-s + (−0.0398 − 0.0289i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78068 + 0.997139i\)
\(L(\frac12)\) \(\approx\) \(1.78068 + 0.997139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.148 - 0.457i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.29 + 0.941i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (0.389 + 0.282i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.48 - 4.56i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.773 - 2.37i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.65 + 3.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.43T + 23T^{2} \)
29 \( 1 + (7.29 + 5.29i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.46 + 7.58i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.20 + 3.05i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.83 + 4.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.53T + 43T^{2} \)
47 \( 1 + (7.76 - 5.64i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.88 + 5.80i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.23 + 1.62i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.24 - 3.82i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 7.60T + 67T^{2} \)
71 \( 1 + (0.855 + 2.63i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.04 - 2.94i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.12 - 3.44i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.529 + 1.63i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (1.12 - 3.45i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.10139640876720421511793988395, −9.671175289071336318898632301717, −8.952015447959265084971810951762, −7.88021872790583255628046044349, −7.32133803523724533444592295452, −6.66489388683132011956673803079, −5.53321017384669481113843678884, −3.93321782168613006617412015504, −2.76615393699004584777514559074, −1.98052681094520077457353075456, 1.15905538086359715335573741695, 2.81402540036020946971098493003, 3.39298871707653363773889540171, 5.04207742436994184470214643686, 5.78626330469889285565075135231, 7.03031297789724588864072573345, 7.963114825226256260274413200824, 9.083706563557470737540020483157, 9.639937781005375155420267109385, 10.34296150937891060021229793519

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