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Properties

Label 2-1035-15.2-c1-0-25
Degree $2$
Conductor $1035$
Sign $0.485 + 0.874i$
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  + (−0.637 + 0.637i)2-s + 1.18i·4-s + (−2.21 + 0.275i)5-s + (1.16 + 1.16i)7-s + (−2.03 − 2.03i)8-s + (1.23 − 1.58i)10-s − 3.46i·11-s + (−3.87 + 3.87i)13-s − 1.49·14-s + 0.215·16-s + (0.633 − 0.633i)17-s − 4.12i·19-s + (−0.326 − 2.63i)20-s + (2.20 + 2.20i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.450 + 0.450i)2-s + 0.593i·4-s + (−0.992 + 0.122i)5-s + (0.442 + 0.442i)7-s + (−0.718 − 0.718i)8-s + (0.391 − 0.502i)10-s − 1.04i·11-s + (−1.07 + 1.07i)13-s − 0.398·14-s + 0.0538·16-s + (0.153 − 0.153i)17-s − 0.947i·19-s + (−0.0730 − 0.589i)20-s + (0.470 + 0.470i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.485 + 0.874i$
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: \((2,\ 1035,\ (\ :1/2),\ 0.485 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4030746759\)
\(L(\frac12)\) \(\approx\) \(0.4030746759\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.275i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.637 - 0.637i)T - 2iT^{2} \)
7 \( 1 + (-1.16 - 1.16i)T + 7iT^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (3.87 - 3.87i)T - 13iT^{2} \)
17 \( 1 + (-0.633 + 0.633i)T - 17iT^{2} \)
19 \( 1 + 4.12iT - 19T^{2} \)
29 \( 1 - 1.24T + 29T^{2} \)
31 \( 1 + 9.87T + 31T^{2} \)
37 \( 1 + (0.527 + 0.527i)T + 37iT^{2} \)
41 \( 1 + 3.76iT - 41T^{2} \)
43 \( 1 + (-8.37 + 8.37i)T - 43iT^{2} \)
47 \( 1 + (2.82 - 2.82i)T - 47iT^{2} \)
53 \( 1 + (-5.48 - 5.48i)T + 53iT^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + (-5.00 - 5.00i)T + 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (-10.6 + 10.6i)T - 73iT^{2} \)
79 \( 1 - 0.454iT - 79T^{2} \)
83 \( 1 + (3.40 + 3.40i)T + 83iT^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 + (11.0 + 11.0i)T + 97iT^{2} \)
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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

−9.340057072422370996972906320229, −8.887189067278979877116412051963, −8.131301896803273836545031640162, −7.30119902188943771949550909850, −6.83029686730865560049897712672, −5.53717892117879529458132491290, −4.42275823281217884133108870544, −3.52462281583320014520958732698, −2.43689895506122821281866651705, −0.23045705150771715970405811004, 1.23383905380257304927258423641, 2.51757148763105507285729772753, 3.84326162745155134946305064193, 4.85793119146416807002298076625, 5.59015848235587900138977696571, 6.96632771449291668156731154547, 7.73402172291342157455648726829, 8.324928542375407213738597060139, 9.540096862263815281173366130527, 10.01826218075554638853829840015

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