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Properties

Label 2-1014-13.3-c1-0-7
Degree $2$
Conductor $1014$
Sign $0.379 - 0.925i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 3.15·5-s + (0.499 − 0.866i)6-s + (−2.34 + 4.06i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.57 − 2.73i)10-s + (0.0685 + 0.118i)11-s − 0.999·12-s + 4.69·14-s + (1.57 + 2.73i)15-s + (−0.5 − 0.866i)16-s + (2.80 − 4.85i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.41·5-s + (0.204 − 0.353i)6-s + (−0.886 + 1.53i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.499 − 0.865i)10-s + (0.0206 + 0.0357i)11-s − 0.288·12-s + 1.25·14-s + (0.407 + 0.706i)15-s + (−0.125 − 0.216i)16-s + (0.679 − 1.17i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.379 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.520634311\)
\(L(\frac12)\) \(\approx\) \(1.520634311\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good5 \( 1 - 3.15T + 5T^{2} \)
7 \( 1 + (2.34 - 4.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0685 - 0.118i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.80 + 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.49 - 4.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.04 - 5.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.425 - 0.736i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.23T + 31T^{2} \)
37 \( 1 + (-5.85 - 10.1i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.13 + 3.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.04 + 1.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.98T + 47T^{2} \)
53 \( 1 + 1.82T + 53T^{2} \)
59 \( 1 + (2.94 - 5.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.35 + 4.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0489 - 0.0847i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.32T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 9.85T + 83T^{2} \)
89 \( 1 + (-8.54 - 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.06 + 1.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.813636303187835028467069058286, −9.420422928244829225490610136714, −8.976603749490547329611745585632, −7.899314541008838752812632579162, −6.55670357306330786480573330458, −5.67767070765669742173188734997, −5.08341954911166875047811395665, −3.41784823087473410864147372931, −2.69477409647513252205481182198, −1.75152993023529725966261822259, 0.75008771817194603582927754165, 2.04066713534680500980192909593, 3.42898006180735019344011982164, 4.63672044563544960838973131800, 5.94556602532545238594518635178, 6.46214510309709154812687939336, 7.15146864969184606656506113632, 8.029818324127840567228580600390, 9.111604207503472447235997837298, 9.633635568102869159984827091636

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