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Properties

Label 2-1035-15.2-c1-0-20
Degree $2$
Conductor $1035$
Sign $0.678 - 0.734i$
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  + (0.842 − 0.842i)2-s + 0.579i·4-s + (0.954 + 2.02i)5-s + (2.82 + 2.82i)7-s + (2.17 + 2.17i)8-s + (2.50 + 0.899i)10-s − 4.87i·11-s + (−2.18 + 2.18i)13-s + 4.75·14-s + 2.50·16-s + (−3.29 + 3.29i)17-s − 2.98i·19-s + (−1.17 + 0.552i)20-s + (−4.10 − 4.10i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (0.595 − 0.595i)2-s + 0.289i·4-s + (0.426 + 0.904i)5-s + (1.06 + 1.06i)7-s + (0.768 + 0.768i)8-s + (0.793 + 0.284i)10-s − 1.46i·11-s + (−0.606 + 0.606i)13-s + 1.27·14-s + 0.626·16-s + (−0.798 + 0.798i)17-s − 0.685i·19-s + (−0.261 + 0.123i)20-s + (−0.875 − 0.875i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(8.26451\)
Root analytic conductor: \(2.87480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.582514805\)
\(L(\frac12)\) \(\approx\) \(2.582514805\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.954 - 2.02i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.842 + 0.842i)T - 2iT^{2} \)
7 \( 1 + (-2.82 - 2.82i)T + 7iT^{2} \)
11 \( 1 + 4.87iT - 11T^{2} \)
13 \( 1 + (2.18 - 2.18i)T - 13iT^{2} \)
17 \( 1 + (3.29 - 3.29i)T - 17iT^{2} \)
19 \( 1 + 2.98iT - 19T^{2} \)
29 \( 1 - 2.36T + 29T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + (-5.89 - 5.89i)T + 37iT^{2} \)
41 \( 1 + 8.87iT - 41T^{2} \)
43 \( 1 + (1.71 - 1.71i)T - 43iT^{2} \)
47 \( 1 + (-8.69 + 8.69i)T - 47iT^{2} \)
53 \( 1 + (2.30 + 2.30i)T + 53iT^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 7.72T + 61T^{2} \)
67 \( 1 + (3.80 + 3.80i)T + 67iT^{2} \)
71 \( 1 + 0.183iT - 71T^{2} \)
73 \( 1 + (1.77 - 1.77i)T - 73iT^{2} \)
79 \( 1 + 8.24iT - 79T^{2} \)
83 \( 1 + (-4.32 - 4.32i)T + 83iT^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + (-12.9 - 12.9i)T + 97iT^{2} \)
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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

−10.43331428997282302419227760048, −9.025232639387392135910828431596, −8.517014045012046789559854765639, −7.60561999735659989732537100863, −6.50810330282700708638676459659, −5.61354009965129348893344514430, −4.74883418533328147616451564441, −3.64393861296831280155639737190, −2.59146187610163136795898728245, −1.98745383348008554315153061485, 1.02336290914827628476655625083, 2.10321405407383981299953393171, 4.22779927032434671710992766237, 4.63366587200553443762196871759, 5.28912319468864110790624706585, 6.33299327539356876937729442255, 7.50299693272457552812170282485, 7.66165061959234048138692427593, 9.081675441526133876557023476878, 9.911171834185906494217061307297

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