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Properties

Label 2-605-11.5-c1-0-14
Degree $2$
Conductor $605$
Sign $-0.0219 - 0.999i$
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  + (0.309 + 0.951i)2-s + (2.42 + 1.76i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.927 + 2.85i)6-s + (−2.42 + 1.76i)7-s + (2.42 + 1.76i)8-s + (1.85 + 5.70i)9-s + 0.999·10-s + 3.00·12-s + (−1.23 − 3.80i)13-s + (−2.42 − 1.76i)14-s + (2.42 − 1.76i)15-s + (−0.309 + 0.951i)16-s + (−4.85 + 3.52i)18-s + (3.23 + 2.35i)19-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (1.40 + 1.01i)3-s + (0.404 − 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.378 + 1.16i)6-s + (−0.917 + 0.666i)7-s + (0.858 + 0.623i)8-s + (0.618 + 1.90i)9-s + 0.316·10-s + 0.866·12-s + (−0.342 − 1.05i)13-s + (−0.648 − 0.471i)14-s + (0.626 − 0.455i)15-s + (−0.0772 + 0.237i)16-s + (−1.14 + 0.831i)18-s + (0.742 + 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0219 - 0.999i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96065 + 2.00424i\)
\(L(\frac12)\) \(\approx\) \(1.96065 + 2.00424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-2.42 - 1.76i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (2.42 - 1.76i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.23 + 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.23 - 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (-4.85 + 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.618 + 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.47 + 4.70i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.04 + 2.93i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-2.42 - 1.76i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.23 - 3.80i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.61 + 1.17i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.39 + 10.4i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + (-0.618 + 1.90i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.47 - 4.70i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.09 + 9.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.23 - 3.80i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (2.47 + 7.60i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.29083784713970957602880595481, −9.993566606569989122163205727731, −9.152753339938196458833019820211, −8.150200793132630152594076203025, −7.64343735898290953660201753794, −6.16547955119399647942857928857, −5.43611200995842551555214591546, −4.32204418098394331230589003818, −3.14380066441807536764370917132, −2.20741332072551306284384933886, 1.49866035777628387405669385230, 2.58991243857458834174726776844, 3.28640092794010035492247148150, 4.21097683628254417425972646215, 6.45367993122772090489773046504, 6.96791076227146943077935863078, 7.62238297446304699059498245615, 8.620618876704277126421666619044, 9.730263200373824823348349757494, 10.22745860016420849388414066673

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