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Properties

Label 2-605-11.3-c1-0-1
Degree $2$
Conductor $605$
Sign $-0.927 + 0.374i$
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  + (−1.82 + 1.32i)2-s + (0.240 + 0.739i)3-s + (0.957 − 2.94i)4-s + (−0.809 − 0.587i)5-s + (−1.42 − 1.03i)6-s + (0.0383 − 0.117i)7-s + (0.766 + 2.35i)8-s + (1.93 − 1.40i)9-s + 2.25·10-s + 2.40·12-s + (−4.44 + 3.22i)13-s + (0.0865 + 0.266i)14-s + (0.240 − 0.739i)15-s + (0.482 + 0.350i)16-s + (−0.420 − 0.305i)17-s + (−1.67 + 5.14i)18-s + ⋯
L(s)  = 1  + (−1.29 + 0.938i)2-s + (0.138 + 0.426i)3-s + (0.478 − 1.47i)4-s + (−0.361 − 0.262i)5-s + (−0.579 − 0.421i)6-s + (0.0144 − 0.0445i)7-s + (0.271 + 0.834i)8-s + (0.646 − 0.469i)9-s + 0.714·10-s + 0.695·12-s + (−1.23 + 0.895i)13-s + (0.0231 + 0.0711i)14-s + (0.0620 − 0.190i)15-s + (0.120 + 0.0875i)16-s + (−0.101 − 0.0740i)17-s + (−0.393 + 1.21i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0567848 - 0.292351i\)
\(L(\frac12)\) \(\approx\) \(0.0567848 - 0.292351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (1.82 - 1.32i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.240 - 0.739i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.0383 + 0.117i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.44 - 3.22i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.420 + 0.305i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.976 - 3.00i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.92T + 23T^{2} \)
29 \( 1 + (1.25 - 3.87i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.70 - 4.14i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.71 - 8.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.33 + 7.18i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (-0.141 - 0.434i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.0287 - 0.0208i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.69 - 5.20i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.22 + 4.51i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 + (9.92 + 7.21i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.63 + 8.10i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.07 - 3.68i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.506 + 0.368i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + (-1.48 + 1.07i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.61017617312919834206416308051, −9.970533387497339877648624229408, −9.279861423798987170437696349718, −8.632818866494663953347013320567, −7.54574357522200059376631992551, −7.08146623760194864614277192858, −6.01384333379406722046001489156, −4.76358335333300497904137938399, −3.68086051413668763614459375436, −1.63964189851892082440002864301, 0.24444803778015019238059081238, 1.92764647494473629218231544609, 2.76322330985091459721988734084, 4.17616013334000436309583217884, 5.61752567011621458659358074546, 7.22048010000649867982252311469, 7.64795636944209801598918419822, 8.412705507476128754007499402941, 9.547587404123430614551829208564, 10.10538560353208029912312179344

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