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Properties

Label 2-605-55.32-c1-0-18
Degree $2$
Conductor $605$
Sign $-0.103 - 0.994i$
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  + (−1.07 + 1.07i)2-s + (0.563 − 0.563i)3-s − 0.310i·4-s + (2.23 + 0.152i)5-s + 1.21i·6-s + (0.135 − 0.135i)7-s + (−1.81 − 1.81i)8-s + 2.36i·9-s + (−2.56 + 2.23i)10-s + (−0.175 − 0.175i)12-s + (2.18 + 2.18i)13-s + 0.291i·14-s + (1.34 − 1.17i)15-s + 4.52·16-s + (−2.62 + 2.62i)17-s + (−2.54 − 2.54i)18-s + ⋯
L(s)  = 1  + (−0.760 + 0.760i)2-s + (0.325 − 0.325i)3-s − 0.155i·4-s + (0.997 + 0.0684i)5-s + 0.494i·6-s + (0.0511 − 0.0511i)7-s + (−0.642 − 0.642i)8-s + 0.788i·9-s + (−0.810 + 0.706i)10-s + (−0.0505 − 0.0505i)12-s + (0.607 + 0.607i)13-s + 0.0777i·14-s + (0.346 − 0.302i)15-s + 1.13·16-s + (−0.635 + 0.635i)17-s + (−0.599 − 0.599i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.103 - 0.994i$
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: \((2,\ 605,\ (\ :1/2),\ -0.103 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835565 + 0.926574i\)
\(L(\frac12)\) \(\approx\) \(0.835565 + 0.926574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.152i)T \)
11 \( 1 \)
good2 \( 1 + (1.07 - 1.07i)T - 2iT^{2} \)
3 \( 1 + (-0.563 + 0.563i)T - 3iT^{2} \)
7 \( 1 + (-0.135 + 0.135i)T - 7iT^{2} \)
13 \( 1 + (-2.18 - 2.18i)T + 13iT^{2} \)
17 \( 1 + (2.62 - 2.62i)T - 17iT^{2} \)
19 \( 1 + 0.743T + 19T^{2} \)
23 \( 1 + (1.14 - 1.14i)T - 23iT^{2} \)
29 \( 1 - 9.54T + 29T^{2} \)
31 \( 1 - 0.350T + 31T^{2} \)
37 \( 1 + (3.82 + 3.82i)T + 37iT^{2} \)
41 \( 1 - 6.69iT - 41T^{2} \)
43 \( 1 + (-3.72 - 3.72i)T + 43iT^{2} \)
47 \( 1 + (8.74 + 8.74i)T + 47iT^{2} \)
53 \( 1 + (-6.38 + 6.38i)T - 53iT^{2} \)
59 \( 1 - 9.62iT - 59T^{2} \)
61 \( 1 - 5.89iT - 61T^{2} \)
67 \( 1 + (-4.13 - 4.13i)T + 67iT^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (1.69 + 1.69i)T + 73iT^{2} \)
79 \( 1 - 0.670T + 79T^{2} \)
83 \( 1 + (11.7 + 11.7i)T + 83iT^{2} \)
89 \( 1 + 7.92iT - 89T^{2} \)
97 \( 1 + (-0.975 - 0.975i)T + 97iT^{2} \)
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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

−10.53113397636508369483156709660, −9.884683020749545664300997083765, −8.711039163449995235793918838627, −8.509627656144806956337518848295, −7.34015544923597888539203877477, −6.58245339113147214124788229758, −5.82090100782990132289313890313, −4.44412388325197369562296625135, −2.88602517430350996932932041911, −1.60147374676733394642847466116, 0.925958406123799510895078049144, 2.31091651151302731515105200828, 3.27029193806528343426536010626, 4.81904062881698904638357439738, 5.92571096922656248004183840905, 6.70739287820345608096740704696, 8.383468572921525896660694340304, 8.852509505270998276697949048629, 9.683356583761802155990363251851, 10.24063086057088768034806903991

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