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Properties

Label 2-1035-15.8-c1-0-17
Degree $2$
Conductor $1035$
Sign $-0.995 - 0.0901i$
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  + (1.79 + 1.79i)2-s + 4.42i·4-s + (0.646 + 2.14i)5-s + (−0.345 + 0.345i)7-s + (−4.34 + 4.34i)8-s + (−2.67 + 4.99i)10-s + 0.0944i·11-s + (−0.829 − 0.829i)13-s − 1.23·14-s − 6.71·16-s + (1.32 + 1.32i)17-s + 0.316i·19-s + (−9.46 + 2.86i)20-s + (−0.169 + 0.169i)22-s + (−0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)2-s + 2.21i·4-s + (0.289 + 0.957i)5-s + (−0.130 + 0.130i)7-s + (−1.53 + 1.53i)8-s + (−0.846 + 1.57i)10-s + 0.0284i·11-s + (−0.230 − 0.230i)13-s − 0.330·14-s − 1.67·16-s + (0.322 + 0.322i)17-s + 0.0726i·19-s + (−2.11 + 0.639i)20-s + (−0.0360 + 0.0360i)22-s + (−0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $-0.995 - 0.0901i$
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: \((2,\ 1035,\ (\ :1/2),\ -0.995 - 0.0901i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.058474758\)
\(L(\frac12)\) \(\approx\) \(3.058474758\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.646 - 2.14i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-1.79 - 1.79i)T + 2iT^{2} \)
7 \( 1 + (0.345 - 0.345i)T - 7iT^{2} \)
11 \( 1 - 0.0944iT - 11T^{2} \)
13 \( 1 + (0.829 + 0.829i)T + 13iT^{2} \)
17 \( 1 + (-1.32 - 1.32i)T + 17iT^{2} \)
19 \( 1 - 0.316iT - 19T^{2} \)
29 \( 1 - 4.62T + 29T^{2} \)
31 \( 1 + 6.04T + 31T^{2} \)
37 \( 1 + (-6.84 + 6.84i)T - 37iT^{2} \)
41 \( 1 + 8.77iT - 41T^{2} \)
43 \( 1 + (1.95 + 1.95i)T + 43iT^{2} \)
47 \( 1 + (-4.30 - 4.30i)T + 47iT^{2} \)
53 \( 1 + (-2.54 + 2.54i)T - 53iT^{2} \)
59 \( 1 - 9.28T + 59T^{2} \)
61 \( 1 + 2.16T + 61T^{2} \)
67 \( 1 + (5.18 - 5.18i)T - 67iT^{2} \)
71 \( 1 + 9.33iT - 71T^{2} \)
73 \( 1 + (-5.77 - 5.77i)T + 73iT^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + (-4.23 + 4.23i)T - 83iT^{2} \)
89 \( 1 + 9.90T + 89T^{2} \)
97 \( 1 + (-3.36 + 3.36i)T - 97iT^{2} \)
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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

−10.39815075688797269433157064869, −9.377762443804315223200981709126, −8.279616714866554460012316133795, −7.42189048236382328971378733360, −6.90605708854558113906241539730, −5.92329742206507659499072968524, −5.51161793975689188734088175512, −4.26070729624446564077195169681, −3.44005036192403903557349520658, −2.43007239024193096457838717913, 0.945613056957781708669877538839, 2.08556969540828333980536490965, 3.17010781733388873745102908528, 4.24532290414406799350430173214, 4.89876393840338557585827826571, 5.68113961439000281811006426601, 6.59559489444952834526033037988, 7.971470454853377690439229281666, 9.060130552680662762655590377866, 9.838601936549295656941330770136

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