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Properties

Label 2-1032-1032.899-c0-0-1
Degree $2$
Conductor $1032$
Sign $0.526 + 0.849i$
Analytic cond. $0.515035$
Root an. cond. $0.717659$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.623 + 0.781i)3-s + (−0.222 − 0.974i)4-s + (0.222 + 0.974i)6-s + (−0.900 − 0.433i)8-s + (−0.222 − 0.974i)9-s + (1.52 + 0.347i)11-s + (0.900 + 0.433i)12-s + (−0.900 + 0.433i)16-s + (−0.846 − 1.75i)17-s + (−0.900 − 0.433i)18-s + (1.90 − 0.433i)19-s + (1.22 − 0.974i)22-s + (0.900 − 0.433i)24-s + (0.623 + 0.781i)25-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.623 + 0.781i)3-s + (−0.222 − 0.974i)4-s + (0.222 + 0.974i)6-s + (−0.900 − 0.433i)8-s + (−0.222 − 0.974i)9-s + (1.52 + 0.347i)11-s + (0.900 + 0.433i)12-s + (−0.900 + 0.433i)16-s + (−0.846 − 1.75i)17-s + (−0.900 − 0.433i)18-s + (1.90 − 0.433i)19-s + (1.22 − 0.974i)22-s + (0.900 − 0.433i)24-s + (0.623 + 0.781i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1032\)    =    \(2^{3} \cdot 3 \cdot 43\)
Sign: $0.526 + 0.849i$
Analytic conductor: \(0.515035\)
Root analytic conductor: \(0.717659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1032} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1032,\ (\ :0),\ 0.526 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171751487\)
\(L(\frac12)\) \(\approx\) \(1.171751487\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
good5 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \)
19 \( 1 + (-1.90 + 0.433i)T + (0.900 - 0.433i)T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.222 + 0.974i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.678 + 1.40i)T + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.222 + 0.974i)T^{2} \)
67 \( 1 + (-0.445 - 1.94i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.22 - 0.974i)T + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \)
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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

−9.922130097983981457170669044894, −9.453161910747605625078567562826, −8.951679473139399060753684627583, −7.08375711906711083600785493739, −6.50388532208214332491923933844, −5.25547388677297963854289222778, −4.82392491938190210955464849749, −3.77417488848239479504694940842, −2.92859718540455070527566139739, −1.17521059932786883500957048979, 1.58994357742362585175660918901, 3.25298083693259266772704716617, 4.27345548247117229470506915850, 5.30420554671637413058750318287, 6.27551015953153972734297624213, 6.57355101466705781889054768292, 7.58837640815330499283854973779, 8.376490021320740066050060586447, 9.131731422521368325224355911133, 10.39666410058990380469157688110

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