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Properties

Label 2-605-5.4-c1-0-2
Degree $2$
Conductor $605$
Sign $-0.774 + 0.632i$
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  + 1.93i·2-s + 0.517i·3-s − 1.73·4-s + (−1.73 + 1.41i)5-s − 0.999·6-s + 0.896i·7-s + 0.517i·8-s + 2.73·9-s + (−2.73 − 3.34i)10-s − 0.896i·12-s + 4.24i·13-s − 1.73·14-s + (−0.732 − 0.896i)15-s − 4.46·16-s + 1.03i·17-s + 5.27i·18-s + ⋯
L(s)  = 1  + 1.36i·2-s + 0.298i·3-s − 0.866·4-s + (−0.774 + 0.632i)5-s − 0.408·6-s + 0.338i·7-s + 0.183i·8-s + 0.910·9-s + (−0.863 − 1.05i)10-s − 0.258i·12-s + 1.17i·13-s − 0.462·14-s + (−0.189 − 0.231i)15-s − 1.11·16-s + 0.251i·17-s + 1.24i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365983 - 1.02690i\)
\(L(\frac12)\) \(\approx\) \(0.365983 - 1.02690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.73 - 1.41i)T \)
11 \( 1 \)
good2 \( 1 - 1.93iT - 2T^{2} \)
3 \( 1 - 0.517iT - 3T^{2} \)
7 \( 1 - 0.896iT - 7T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 - 1.03iT - 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + 6.69iT - 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + 4.10iT - 47T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 - 4.73T + 59T^{2} \)
61 \( 1 + 8.26T + 61T^{2} \)
67 \( 1 - 14.9iT - 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 - 5.46T + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 + 0.656iT - 97T^{2} \)
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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

−11.01972397208945786982328248829, −10.35259405051268750661220357001, −9.052862473202416574203539514443, −8.434977885181174537762924384351, −7.42820029238289958177891034397, −6.74812415505625711709484517358, −6.12413636863721053749206466826, −4.63029537401670829393468099158, −4.11628589610190281891351522260, −2.35338357138096273518450029258, 0.61272315266525956304406495685, 1.81202633443824465286063909790, 3.30314056029831086999845595764, 4.11624291806191834729918535260, 5.08471060381130198038423743851, 6.67512372258759353124449506476, 7.59631746833797786297925225052, 8.478913626439602374869626865996, 9.519097690224514053681050373515, 10.37799605590790873746030034799

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