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Properties

Label 2-1014-13.4-c1-0-3
Degree $2$
Conductor $1014$
Sign $0.265 - 0.964i$
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 + 0.267i·5-s + (0.866 − 0.499i)6-s + (−0.633 + 0.366i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.133 − 0.232i)10-s + (4.09 + 2.36i)11-s − 0.999·12-s + 0.732·14-s + (−0.232 − 0.133i)15-s + (−0.5 + 0.866i)16-s + (−1.13 − 1.96i)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 + 0.119i·5-s + (0.353 − 0.204i)6-s + (−0.239 + 0.138i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.0423 − 0.0733i)10-s + (1.23 + 0.713i)11-s − 0.288·12-s + 0.195·14-s + (−0.0599 − 0.0345i)15-s + (−0.125 + 0.216i)16-s + (−0.275 − 0.476i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (823, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9514711019\)
\(L(\frac12)\) \(\approx\) \(0.9514711019\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.267iT - 5T^{2} \)
7 \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.13 + 1.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.09 - 5.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (-9.06 - 5.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.86 + 5.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.83 - 6.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.19iT - 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (6.92 - 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.598 + 1.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.63 - 5.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.09 + 0.633i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (2.19 + 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.757626578009695612471664601470, −9.622092537506284442593700619918, −8.756746670516324385007147501827, −7.65527332168645984456412392348, −6.81489366428869604096207443635, −5.98973270845087938681972425065, −4.72896702418176111503571888232, −3.84559477538275105199807516812, −2.74254367422530931991557324841, −1.30135589303490913090778949103, 0.61935289763040152894160624879, 1.88200274704251013697720762676, 3.39410820782102921621289051727, 4.62181024477718259820167225721, 5.85331076841406113554900825368, 6.49069139393279802468953377967, 7.13949829853331713762736349520, 8.304139991238758996824026773075, 8.738524439555539119248274336853, 9.709850615135934141826922929368

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