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Properties

Label 2-285-285.149-c0-0-0
Degree $2$
Conductor $285$
Sign $0.775 + 0.631i$
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  + (−0.266 − 0.223i)2-s + (0.939 − 0.342i)3-s + (−0.152 − 0.866i)4-s + (−0.173 + 0.984i)5-s + (−0.326 − 0.118i)6-s + (−0.326 + 0.565i)8-s + (0.766 − 0.642i)9-s + (0.266 − 0.223i)10-s + (−0.439 − 0.761i)12-s + (0.173 + 0.984i)15-s + (−0.613 + 0.223i)16-s + (−1.17 − 0.984i)17-s − 0.347·18-s + (−0.939 + 0.342i)19-s + 0.879·20-s + ⋯
L(s)  = 1  + (−0.266 − 0.223i)2-s + (0.939 − 0.342i)3-s + (−0.152 − 0.866i)4-s + (−0.173 + 0.984i)5-s + (−0.326 − 0.118i)6-s + (−0.326 + 0.565i)8-s + (0.766 − 0.642i)9-s + (0.266 − 0.223i)10-s + (−0.439 − 0.761i)12-s + (0.173 + 0.984i)15-s + (−0.613 + 0.223i)16-s + (−1.17 − 0.984i)17-s − 0.347·18-s + (−0.939 + 0.342i)19-s + 0.879·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(0.142233\)
Root analytic conductor: \(0.377138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :0),\ 0.775 + 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8148573823\)
\(L(\frac12)\) \(\approx\) \(0.8148573823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
good2 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−11.75603315119695408329819809884, −10.88619529523012053632790713806, −10.01581810074565373620712024108, −9.192424293361889507874789127198, −8.251512981924208481980874775179, −7.03658884275744746093897978129, −6.30895244133513546097602572285, −4.66319841044178997023146095721, −3.15058520749016528264260621096, −1.96321002090837839465893124632, 2.39672727532387591428564775451, 3.99394552226346928766970156371, 4.57010303344770628017073012941, 6.45494964737171486888241368266, 7.72251377948068648897041098785, 8.525225876765994814203008619051, 8.904818907742826603864928456069, 9.974259575384854705100054076657, 11.21790654826515503560418355681, 12.47999938148472640420213382262

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