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Properties

Label 2-1035-15.8-c1-0-35
Degree $2$
Conductor $1035$
Sign $0.963 - 0.266i$
Analytic cond. $8.26451$
Root an. cond. $2.87480$
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  + (1.65 + 1.65i)2-s + 3.49i·4-s + (−1.35 − 1.77i)5-s + (2.43 − 2.43i)7-s + (−2.47 + 2.47i)8-s + (0.691 − 5.19i)10-s − 4.87i·11-s + (−3.02 − 3.02i)13-s + 8.07·14-s − 1.21·16-s + (4.90 + 4.90i)17-s − 4.20i·19-s + (6.20 − 4.74i)20-s + (8.07 − 8.07i)22-s + (−0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)2-s + 1.74i·4-s + (−0.607 − 0.794i)5-s + (0.921 − 0.921i)7-s + (−0.874 + 0.874i)8-s + (0.218 − 1.64i)10-s − 1.46i·11-s + (−0.839 − 0.839i)13-s + 2.15·14-s − 0.302·16-s + (1.18 + 1.18i)17-s − 0.965i·19-s + (1.38 − 1.06i)20-s + (1.72 − 1.72i)22-s + (−0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.963 - 0.266i$
Analytic conductor: \(8.26451\)
Root analytic conductor: \(2.87480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :1/2),\ 0.963 - 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.857050387\)
\(L(\frac12)\) \(\approx\) \(2.857050387\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.35 + 1.77i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-1.65 - 1.65i)T + 2iT^{2} \)
7 \( 1 + (-2.43 + 2.43i)T - 7iT^{2} \)
11 \( 1 + 4.87iT - 11T^{2} \)
13 \( 1 + (3.02 + 3.02i)T + 13iT^{2} \)
17 \( 1 + (-4.90 - 4.90i)T + 17iT^{2} \)
19 \( 1 + 4.20iT - 19T^{2} \)
29 \( 1 + 5.57T + 29T^{2} \)
31 \( 1 - 7.38T + 31T^{2} \)
37 \( 1 + (-3.41 + 3.41i)T - 37iT^{2} \)
41 \( 1 - 6.52iT - 41T^{2} \)
43 \( 1 + (0.560 + 0.560i)T + 43iT^{2} \)
47 \( 1 + (8.94 + 8.94i)T + 47iT^{2} \)
53 \( 1 + (9.39 - 9.39i)T - 53iT^{2} \)
59 \( 1 + 1.34T + 59T^{2} \)
61 \( 1 - 6.67T + 61T^{2} \)
67 \( 1 + (-2.51 + 2.51i)T - 67iT^{2} \)
71 \( 1 - 5.54iT - 71T^{2} \)
73 \( 1 + (-11.6 - 11.6i)T + 73iT^{2} \)
79 \( 1 + 6.00iT - 79T^{2} \)
83 \( 1 + (-0.475 + 0.475i)T - 83iT^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (2.46 - 2.46i)T - 97iT^{2} \)
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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

−9.982591246462674259813336932530, −8.589164113213697211165881675995, −7.83676575042701828081764578642, −7.71352211542158114837585210413, −6.42508218960802265348613923529, −5.47917164816171369862927508232, −4.91038887794927549495836452061, −4.01487214912115181460941859076, −3.24807534181304178327429166869, −0.954456647050118447746501867042, 1.80401291273546997355667664510, 2.50542443509603520582134994484, 3.54763106232413391169564447820, 4.66673125413323940635610738609, 5.03776022053442827635620100029, 6.26123075914370989477654211559, 7.40424289468652324487412765580, 8.060965868453709546649376926990, 9.692817590877492313722007751757, 9.906904870991022808046917120640

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