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Properties

Label 2-605-11.3-c1-0-24
Degree $2$
Conductor $605$
Sign $0.970 + 0.242i$
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  + (1.09 − 0.796i)2-s + (0.177 + 0.547i)3-s + (−0.0501 + 0.154i)4-s + (0.809 + 0.587i)5-s + (0.631 + 0.458i)6-s + (1.12 − 3.47i)7-s + (0.905 + 2.78i)8-s + (2.15 − 1.56i)9-s + 1.35·10-s − 0.0933·12-s + (−2.29 + 1.66i)13-s + (−1.52 − 4.70i)14-s + (−0.177 + 0.547i)15-s + (2.95 + 2.14i)16-s + (2.98 + 2.17i)17-s + (1.11 − 3.44i)18-s + ⋯
L(s)  = 1  + (0.775 − 0.563i)2-s + (0.102 + 0.315i)3-s + (−0.0250 + 0.0771i)4-s + (0.361 + 0.262i)5-s + (0.257 + 0.187i)6-s + (0.426 − 1.31i)7-s + (0.320 + 0.985i)8-s + (0.719 − 0.522i)9-s + 0.428·10-s − 0.0269·12-s + (−0.635 + 0.461i)13-s + (−0.408 − 1.25i)14-s + (−0.0459 + 0.141i)15-s + (0.738 + 0.536i)16-s + (0.724 + 0.526i)17-s + (0.263 − 0.811i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53250 - 0.311312i\)
\(L(\frac12)\) \(\approx\) \(2.53250 - 0.311312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-1.09 + 0.796i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.177 - 0.547i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-1.12 + 3.47i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.29 - 1.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.98 - 2.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.0293 - 0.0904i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + (-2.08 + 6.42i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.48 - 3.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.04 + 9.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.57 - 7.91i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 + (0.687 + 2.11i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.42 - 1.75i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.62 - 8.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.86 + 4.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (6.71 + 4.88i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.407 + 1.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.2 - 8.15i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.61 + 6.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-3.50 + 2.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.65948712126384385241084697113, −10.03451103688859295204022683557, −9.064456989534793561494320901265, −7.74998818043972677221065459409, −7.18841215272628241479336145181, −5.87093311506560080818596244313, −4.54810827931948857519636946060, −4.10859035886999789234462141083, −3.04287723742262772834507340079, −1.55727474106292716196062021165, 1.51579959284313294205641896248, 2.86649346233235674369881448385, 4.53763806786535294748390340695, 5.25740624906330503541276577750, 5.87114818973755173356005597198, 7.03904491545502420892169484670, 7.80898091548775440959500955985, 8.942219747841255649506526372844, 9.757520670079980577633739972416, 10.59115441216198153057845301370

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