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Properties

Label 2-605-11.5-c1-0-33
Degree $2$
Conductor $605$
Sign $-0.944 + 0.329i$
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  + (0.0288 + 0.0889i)2-s + (−1.18 − 0.862i)3-s + (1.61 − 1.17i)4-s + (0.309 − 0.951i)5-s + (0.0423 − 0.130i)6-s + (−3.66 + 2.65i)7-s + (0.301 + 0.219i)8-s + (−0.262 − 0.807i)9-s + 0.0935·10-s − 2.92·12-s + (−0.353 − 1.08i)13-s + (−0.342 − 0.248i)14-s + (−1.18 + 0.862i)15-s + (1.21 − 3.75i)16-s + (1.04 − 3.20i)17-s + (0.0642 − 0.0466i)18-s + ⋯
L(s)  = 1  + (0.0204 + 0.0628i)2-s + (−0.685 − 0.497i)3-s + (0.805 − 0.585i)4-s + (0.138 − 0.425i)5-s + (0.0173 − 0.0532i)6-s + (−1.38 + 1.00i)7-s + (0.106 + 0.0775i)8-s + (−0.0874 − 0.269i)9-s + 0.0295·10-s − 0.843·12-s + (−0.0979 − 0.301i)13-s + (−0.0914 − 0.0664i)14-s + (−0.306 + 0.222i)15-s + (0.304 − 0.938i)16-s + (0.252 − 0.778i)17-s + (0.0151 − 0.0109i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.944 + 0.329i$
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: \((2,\ 605,\ (\ :1/2),\ -0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.110434 - 0.650866i\)
\(L(\frac12)\) \(\approx\) \(0.110434 - 0.650866i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0288 - 0.0889i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.18 + 0.862i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (3.66 - 2.65i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.353 + 1.08i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.04 + 3.20i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.92 + 3.57i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.45T + 23T^{2} \)
29 \( 1 + (2.68 - 1.95i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.553 - 1.70i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.20 - 0.875i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.41 - 1.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.263T + 43T^{2} \)
47 \( 1 + (5.60 + 4.07i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.444 + 1.36i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.71 + 4.15i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.773 + 2.37i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.516T + 67T^{2} \)
71 \( 1 + (3.31 - 10.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.59 + 3.33i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.49 + 10.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.38 + 4.26i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (1.03 + 3.19i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.20832551294418600038658735009, −9.511234916181681015672748946789, −8.648109726465284866800539721142, −7.21970443670798767612350429608, −6.42682955420149993732519686816, −5.94461462116131046485288404634, −5.09225098930979359348782446226, −3.24109342093974909557958607225, −2.09905235721235898811180792404, −0.35554581291674849780298409028, 2.18120967046739692917107311782, 3.56615378708815446646978775221, 4.19386031593854187078001167991, 5.98138033131498349519178603469, 6.36262661725377223746454277441, 7.38148672899649108471197467909, 8.218259009536080305445749508710, 9.731736861221768495563048271116, 10.37686343034843785743711268286, 10.81772665950290990875425500542

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