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Properties

Label 2-605-55.9-c1-0-45
Degree $2$
Conductor $605$
Sign $-0.990 + 0.133i$
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 + (−2.19 + 0.404i)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 + (4.24 + 4.03i)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 + (−0.983 + 0.180i)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 + (1.09 + 1.04i)15-s + (−0.293 − 0.903i)16-s + (−0.769 + 0.250i)17-s + (−1.08 + 1.49i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.990 + 0.133i$
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: \((2,\ 605,\ (\ :1/2),\ -0.990 + 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0288587 - 0.429073i\)
\(L(\frac12)\) \(\approx\) \(0.0288587 - 0.429073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 - 0.404i)T \)
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

−10.73581596065878706323174801013, −9.286635278461811251290985096280, −7.79762055128482323780106139820, −7.32038280046995499739631400083, −6.45265291420850079370398827164, −5.75568291124081107136761996985, −4.79938309834868013417169853641, −3.75176658358508709068071708388, −2.36940215871433192511090027930, −0.15917955204349557549660927992, 2.71605516523722588062435970547, 3.92843104226726040827920796740, 4.54189709065933806426966347012, 5.00000036841062183412315387644, 6.19039638015040849360582051302, 7.19243617327662184281278866856, 8.622514210340608515005949798162, 9.621633196295322129343582996868, 10.63413023782450771895183080829, 11.26251841013727652577831483115

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