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Properties

Label 2-605-55.14-c1-0-12
Degree $2$
Conductor $605$
Sign $0.967 - 0.252i$
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  + (−0.465 − 0.640i)2-s + (−2.40 + 0.780i)3-s + (0.424 − 1.30i)4-s + (2.04 + 0.904i)5-s + (1.61 + 1.17i)6-s + (3.29 + 1.07i)7-s + (−2.54 + 0.825i)8-s + (2.72 − 1.98i)9-s + (−0.372 − 1.73i)10-s + 3.46i·12-s + (−0.848 − 2.61i)14-s + (−5.61 − 0.577i)15-s + (−0.507 − 0.368i)16-s + (−2.96 + 4.08i)17-s + (−2.54 − 0.825i)18-s + (1.23 + 3.80i)19-s + ⋯
L(s)  = 1  + (−0.329 − 0.453i)2-s + (−1.38 + 0.450i)3-s + (0.212 − 0.652i)4-s + (0.914 + 0.404i)5-s + (0.660 + 0.479i)6-s + (1.24 + 0.404i)7-s + (−0.898 + 0.291i)8-s + (0.909 − 0.660i)9-s + (−0.117 − 0.547i)10-s + 0.999i·12-s + (−0.226 − 0.697i)14-s + (−1.44 − 0.149i)15-s + (−0.126 − 0.0922i)16-s + (−0.719 + 0.990i)17-s + (−0.598 − 0.194i)18-s + (0.283 + 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.967 - 0.252i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01113 + 0.129975i\)
\(L(\frac12)\) \(\approx\) \(1.01113 + 0.129975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.04 - 0.904i)T \)
11 \( 1 \)
good2 \( 1 + (0.465 + 0.640i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.40 - 0.780i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-3.29 - 1.07i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.96 - 4.08i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.23 - 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.52iT - 23T^{2} \)
29 \( 1 + (0.848 - 2.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.91 + 1.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-10.4 - 3.41i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.848 - 2.61i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-6.30 + 2.04i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.86 - 2.56i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.502 + 1.54i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.69 - 6.31i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.644iT - 67T^{2} \)
71 \( 1 + (5.75 + 4.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.58 - 2.14i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.3 - 7.49i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.89 + 5.36i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + (2.41 + 3.32i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.61563431253657120488249902754, −10.25462087790577321781954310116, −9.283366922268974654752447802549, −8.253941161678270559696123446568, −6.70699311529451380981474921320, −5.89927292957348857381083818748, −5.43240148419768411854398054756, −4.41642297986608694249428693626, −2.39382035894500244118004157514, −1.30645081002760085289416572556, 0.864458140676113926479147847971, 2.40303503296913315369193157958, 4.45199238606057827735068106259, 5.26548357082301069985415015738, 6.17574333419783934846218000717, 7.02133243406003894217526975467, 7.72926192862580551228969856766, 8.803944225249063513743818166943, 9.655885330554190805809231926883, 10.98294268451322693724181108463

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