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Properties

Label 2-1014-13.10-c1-0-3
Degree $2$
Conductor $1014$
Sign $0.0502 - 0.998i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s − 0.356i·5-s + (0.866 + 0.499i)6-s + (−3.50 − 2.02i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.178 + 0.309i)10-s + (−0.789 + 0.455i)11-s − 0.999·12-s + 4.04·14-s + (−0.309 + 0.178i)15-s + (−0.5 − 0.866i)16-s + (−1.04 + 1.81i)17-s − 0.999i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s − 0.159i·5-s + (0.353 + 0.204i)6-s + (−1.32 − 0.765i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0564 + 0.0977i)10-s + (−0.238 + 0.137i)11-s − 0.288·12-s + 1.08·14-s + (−0.0798 + 0.0460i)15-s + (−0.125 − 0.216i)16-s + (−0.254 + 0.440i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0502 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0502 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.0502 - 0.998i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.0502 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4301766304\)
\(L(\frac12)\) \(\approx\) \(0.4301766304\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 + 0.356iT - 5T^{2} \)
7 \( 1 + (3.50 + 2.02i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.789 - 0.455i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.04 - 1.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.31 + 2.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.24 - 7.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.25 + 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.7iT - 31T^{2} \)
37 \( 1 + (0.533 - 0.307i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.58 + 3.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.13 - 5.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.78iT - 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (-5.23 - 3.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.55 + 2.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.7 - 6.78i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.94 - 5.74i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.533iT - 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 6.49iT - 83T^{2} \)
89 \( 1 + (5.62 - 3.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.69 + 0.980i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.14231993925206630167823292436, −9.267056796801374048505985725908, −8.557447432174186700812093674122, −7.41199093080547497691808794421, −6.92207848216399021611668349632, −6.19172560537613798131504567723, −5.19187412812582976448421349542, −3.90689688348332723417179370618, −2.65673302535476133473879056804, −1.10294719047501977760319502441, 0.28378706893365250370537829771, 2.38646054006791625134738681855, 3.18416459025987822012642301842, 4.30720279750536168120036685525, 5.56869129747018106338290957323, 6.41781110439520731826826887766, 7.13684739303227940632037663448, 8.496451460307831857298095599641, 9.026136865664928654476214599875, 9.750160808846456421028555856995

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