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Properties

Label 2-285-5.4-c1-0-3
Degree $2$
Conductor $285$
Sign $-0.637 - 0.770i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + 1.57i·2-s − i·3-s − 0.472·4-s + (−1.72 + 1.42i)5-s + 1.57·6-s + 1.87i·7-s + 2.40i·8-s − 9-s + (−2.24 − 2.70i)10-s + 1.92·11-s + 0.472i·12-s − 0.248i·13-s − 2.95·14-s + (1.42 + 1.72i)15-s − 4.72·16-s + 7.06i·17-s + ⋯
L(s)  = 1  + 1.11i·2-s − 0.577i·3-s − 0.236·4-s + (−0.770 + 0.637i)5-s + 0.641·6-s + 0.710i·7-s + 0.849i·8-s − 0.333·9-s + (−0.708 − 0.856i)10-s + 0.579·11-s + 0.136i·12-s − 0.0688i·13-s − 0.789·14-s + (0.367 + 0.444i)15-s − 1.18·16-s + 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489202 + 1.03920i\)
\(L(\frac12)\) \(\approx\) \(0.489202 + 1.03920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (1.72 - 1.42i)T \)
19 \( 1 + T \)
good2 \( 1 - 1.57iT - 2T^{2} \)
7 \( 1 - 1.87iT - 7T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 + 0.248iT - 13T^{2} \)
17 \( 1 - 7.06iT - 17T^{2} \)
23 \( 1 - 1.20iT - 23T^{2} \)
29 \( 1 + 2.58T + 29T^{2} \)
31 \( 1 - 8.69T + 31T^{2} \)
37 \( 1 + 7.86iT - 37T^{2} \)
41 \( 1 - 1.52T + 41T^{2} \)
43 \( 1 + 4.15iT - 43T^{2} \)
47 \( 1 + 5.96iT - 47T^{2} \)
53 \( 1 + 7.11iT - 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 5.87T + 61T^{2} \)
67 \( 1 - 9.53iT - 67T^{2} \)
71 \( 1 + 9.53T + 71T^{2} \)
73 \( 1 - 6.69iT - 73T^{2} \)
79 \( 1 - 0.348T + 79T^{2} \)
83 \( 1 + 2.41iT - 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + 7.70iT - 97T^{2} \)
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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

−12.05858906655738792468277621320, −11.44692876681363067788317592798, −10.38795465019468433306039658413, −8.731868143550460534889896869102, −8.181253231480526813303724259896, −7.18191392964083658408645777829, −6.41312355933328131699971235270, −5.59408960206955577928236281694, −3.92057786202815758025393013048, −2.28683768792493306787600495323, 0.906408484495271902742378628440, 2.91338347050412593072226108311, 4.04898612555584046066121514780, 4.79755359438381268566462160021, 6.62850952737287703030762661423, 7.71575842359589625455890343083, 9.009927870208188780762121730791, 9.759688326978327804950528745074, 10.69073821733730298692643671330, 11.62434102930244809550214700094

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