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Properties

Label 2-285-285.74-c0-0-1
Degree $2$
Conductor $285$
Sign $-0.999 - 0.0158i$
Analytic cond. $0.142233$
Root an. cond. $0.377138$
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  + (0.326 − 1.85i)2-s + (−0.766 − 0.642i)3-s + (−2.37 − 0.866i)4-s + (0.939 − 0.342i)5-s + (−1.43 + 1.20i)6-s + (−1.43 + 2.49i)8-s + (0.173 + 0.984i)9-s + (−0.326 − 1.85i)10-s + (1.26 + 2.19i)12-s + (−0.939 − 0.342i)15-s + (2.20 + 1.85i)16-s + (−0.0603 + 0.342i)17-s + 1.87·18-s + (0.766 + 0.642i)19-s − 2.53·20-s + ⋯
L(s)  = 1  + (0.326 − 1.85i)2-s + (−0.766 − 0.642i)3-s + (−2.37 − 0.866i)4-s + (0.939 − 0.342i)5-s + (−1.43 + 1.20i)6-s + (−1.43 + 2.49i)8-s + (0.173 + 0.984i)9-s + (−0.326 − 1.85i)10-s + (1.26 + 2.19i)12-s + (−0.939 − 0.342i)15-s + (2.20 + 1.85i)16-s + (−0.0603 + 0.342i)17-s + 1.87·18-s + (0.766 + 0.642i)19-s − 2.53·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.999 - 0.0158i$
Analytic conductor: \(0.142233\)
Root analytic conductor: \(0.377138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :0),\ -0.999 - 0.0158i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7047309211\)
\(L(\frac12)\) \(\approx\) \(0.7047309211\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
good2 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−11.67322643407461963624132124599, −10.80870098537646963348993472524, −10.04108727890574291043717868497, −9.256206768155591362586935459890, −7.951689750095745451714574207522, −6.11329808608514073725027894814, −5.34548096444707007130824244582, −4.18344275264860352061814683263, −2.47248564917661304286165415656, −1.38160686124600738178473975199, 3.61754761610011142657138729550, 5.02055399347997694546157077876, 5.54275312247184961434295165470, 6.55279198210966454229377746983, 7.22332758333666695975112602426, 8.689639792739099936211362623496, 9.507221998284358403940451902013, 10.31683401187425220392662733548, 11.70106634436632194847002676249, 12.87694785086210605564491335775

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