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Properties

Label 2-605-55.4-c1-0-29
Degree $2$
Conductor $605$
Sign $0.985 - 0.166i$
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 + (1.49 − 1.66i)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 + (4.88 − 2.83i)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 + (0.667 − 0.744i)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 + (1.26 − 0.731i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.985 - 0.166i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.985 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90328 + 0.244077i\)
\(L(\frac12)\) \(\approx\) \(2.90328 + 0.244077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.49 + 1.66i)T \)
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

−10.30853958771232208050382207102, −9.827661906205891422340922926926, −8.761109542089492011309137534199, −8.508536211464847437525118367460, −7.27604500486403685732764595491, −6.10725693544490536011023900889, −4.78314135711276581566727811999, −3.66406657796062065458169352510, −2.98275004718639751187791432502, −1.99646425150154740001297250200, 1.60445932126713260747907466429, 2.87666288599206460332784501004, 3.62073457586457334618461280118, 5.35003865354789266339006552724, 6.30303031360994881739354117346, 7.10286836362465341082124875453, 7.61346760183573219134372769123, 9.005219699760261078111940416481, 9.921087867512814853755406483001, 10.08998725717891766013088792995

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