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Properties

Label 2-605-55.9-c1-0-18
Degree $2$
Conductor $605$
Sign $0.921 - 0.388i$
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  + (−1.92 − 0.625i)2-s + (1.54 + 2.12i)3-s + (1.69 + 1.23i)4-s + (2.01 − 0.965i)5-s + (−1.63 − 5.04i)6-s + (0.567 − 0.781i)7-s + (−0.113 − 0.156i)8-s + (−1.19 + 3.67i)9-s + (−4.48 + 0.596i)10-s + 5.49i·12-s + (4.30 + 1.39i)13-s + (−1.58 + 1.14i)14-s + (5.15 + 2.78i)15-s + (−1.17 − 3.61i)16-s + (3.17 − 1.03i)17-s + (4.60 − 6.33i)18-s + ⋯
L(s)  = 1  + (−1.36 − 0.442i)2-s + (0.889 + 1.22i)3-s + (0.847 + 0.615i)4-s + (0.901 − 0.431i)5-s + (−0.669 − 2.05i)6-s + (0.214 − 0.295i)7-s + (−0.0400 − 0.0551i)8-s + (−0.398 + 1.22i)9-s + (−1.41 + 0.188i)10-s + 1.58i·12-s + (1.19 + 0.387i)13-s + (−0.422 + 0.307i)14-s + (1.33 + 0.720i)15-s + (−0.293 − 0.903i)16-s + (0.769 − 0.250i)17-s + (1.08 − 1.49i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.921 - 0.388i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22743 + 0.248116i\)
\(L(\frac12)\) \(\approx\) \(1.22743 + 0.248116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.01 + 0.965i)T \)
11 \( 1 \)
good2 \( 1 + (1.92 + 0.625i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.54 - 2.12i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.567 + 0.781i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.30 - 1.39i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.17 + 1.03i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.65 - 1.92i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.36iT - 23T^{2} \)
29 \( 1 + (3.97 + 2.88i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.129 - 0.397i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.72 - 5.12i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.68 + 3.40i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.26iT - 43T^{2} \)
47 \( 1 + (-2.54 - 3.49i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.53 + 0.822i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.19 + 5.95i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.763 - 2.34i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.60iT - 67T^{2} \)
71 \( 1 + (1.68 + 5.18i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.843 - 1.16i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.310 + 0.954i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.03 + 2.28i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-2.87 - 0.932i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.49403318941712478876491747665, −9.671052351187947515293753639160, −9.134696209235206220302618734309, −8.522449942927563556549135123141, −7.78121120971204245385500547595, −6.23951359363013612672599565677, −4.96287861456768112553164687354, −3.91513277748520126163273366214, −2.61104100011539732953519748855, −1.38164916419370291053260005226, 1.25633552932798708442058878479, 2.11365852684418720315803219570, 3.46032407223529196009608440554, 5.66791634078736027301882405267, 6.49415219973163737094341155319, 7.30830863369612383481678939881, 7.997434148411938980709383515168, 8.810171650802304771151740869304, 9.290909474453089296252834197531, 10.40003394510335982785094392263

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