Properties

Label 2-1805-95.83-c0-0-0
Degree $2$
Conductor $1805$
Sign $0.991 + 0.127i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)4-s + (0.5 + 0.866i)5-s + (1 − i)7-s + (0.866 + 0.5i)9-s + (0.499 + 0.866i)16-s + (−1.36 − 0.366i)17-s − 0.999i·20-s + (1.36 − 0.366i)23-s + (−0.499 + 0.866i)25-s + (−1.36 + 0.366i)28-s + (1.36 + 0.366i)35-s + (−0.499 − 0.866i)36-s + (0.366 − 1.36i)43-s + 0.999i·45-s + (0.366 + 1.36i)47-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)4-s + (0.5 + 0.866i)5-s + (1 − i)7-s + (0.866 + 0.5i)9-s + (0.499 + 0.866i)16-s + (−1.36 − 0.366i)17-s − 0.999i·20-s + (1.36 − 0.366i)23-s + (−0.499 + 0.866i)25-s + (−1.36 + 0.366i)28-s + (1.36 + 0.366i)35-s + (−0.499 − 0.866i)36-s + (0.366 − 1.36i)43-s + 0.999i·45-s + (0.366 + 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199789177\)
\(L(\frac12)\) \(\approx\) \(1.199789177\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T^{2} \)
3 \( 1 + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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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

−9.476227354372418581778910711038, −8.805917264168966178844236156431, −7.74197173103607980908265894732, −7.12063549873350007932659631204, −6.35382912818920025163166578765, −5.13860173585919462425595207208, −4.61648928476780265969840442654, −3.80668435486843840396460578373, −2.32380069510543619947206085625, −1.23740987020273788573557037796, 1.28268344158568179228032519218, 2.41986216416759211326095713572, 3.84880483325986339020206104231, 4.72249620430373927388068272272, 5.12415546940069219469263240017, 6.15413903725315429259953379522, 7.22987802873263291797512480868, 8.204610692145135550494078229651, 8.810323944685943917645959433774, 9.193055880707401234820104713577

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