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Properties

Label 2-1035-15.2-c1-0-24
Degree $2$
Conductor $1035$
Sign $0.788 + 0.615i$
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  + (0.246 − 0.246i)2-s + 1.87i·4-s + (−1.93 + 1.11i)5-s + (−1.89 − 1.89i)7-s + (0.954 + 0.954i)8-s + (−0.200 + 0.751i)10-s − 5.44i·11-s + (4.08 − 4.08i)13-s − 0.933·14-s − 3.28·16-s + (2.46 − 2.46i)17-s + 8.65i·19-s + (−2.10 − 3.63i)20-s + (−1.33 − 1.33i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (0.174 − 0.174i)2-s + 0.939i·4-s + (−0.865 + 0.500i)5-s + (−0.716 − 0.716i)7-s + (0.337 + 0.337i)8-s + (−0.0634 + 0.237i)10-s − 1.64i·11-s + (1.13 − 1.13i)13-s − 0.249·14-s − 0.821·16-s + (0.598 − 0.598i)17-s + 1.98i·19-s + (−0.470 − 0.813i)20-s + (−0.285 − 0.285i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.788 + 0.615i$
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: \((2,\ 1035,\ (\ :1/2),\ 0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.313221948\)
\(L(\frac12)\) \(\approx\) \(1.313221948\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 - 1.11i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.246 + 0.246i)T - 2iT^{2} \)
7 \( 1 + (1.89 + 1.89i)T + 7iT^{2} \)
11 \( 1 + 5.44iT - 11T^{2} \)
13 \( 1 + (-4.08 + 4.08i)T - 13iT^{2} \)
17 \( 1 + (-2.46 + 2.46i)T - 17iT^{2} \)
19 \( 1 - 8.65iT - 19T^{2} \)
29 \( 1 + 3.53T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (-5.02 - 5.02i)T + 37iT^{2} \)
41 \( 1 + 6.15iT - 41T^{2} \)
43 \( 1 + (-2.21 + 2.21i)T - 43iT^{2} \)
47 \( 1 + (-6.95 + 6.95i)T - 47iT^{2} \)
53 \( 1 + (0.310 + 0.310i)T + 53iT^{2} \)
59 \( 1 - 4.51T + 59T^{2} \)
61 \( 1 + 4.06T + 61T^{2} \)
67 \( 1 + (0.634 + 0.634i)T + 67iT^{2} \)
71 \( 1 + 5.71iT - 71T^{2} \)
73 \( 1 + (-1.72 + 1.72i)T - 73iT^{2} \)
79 \( 1 + 9.77iT - 79T^{2} \)
83 \( 1 + (6.39 + 6.39i)T + 83iT^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (-1.04 - 1.04i)T + 97iT^{2} \)
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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

−10.18292000881439678726863750790, −8.684276681019075940408148167833, −8.091691261840455525919846146310, −7.58779537590699859628054311738, −6.46591632084712584155400130148, −5.68832667312478780870918096972, −4.05710413157345173290140887026, −3.47599777224356586890171130304, −3.00128309616111432874792215204, −0.67274084516616896506746690719, 1.19329752000788945556976332956, 2.56604290949884912418086941506, 4.18270620530179009429252127673, 4.62946506909373441280479508118, 5.78828998461738257689559637267, 6.61688350734652573468654008784, 7.30767950144344857753818915392, 8.509536593986997287016620829505, 9.373881505364896417268723564388, 9.697208546072810300546475663689

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