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Properties

Label 2-1014-13.3-c1-0-2
Degree $2$
Conductor $1014$
Sign $0.997 - 0.0743i$
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 − 4.04·5-s + (−0.499 + 0.866i)6-s + (0.346 − 0.599i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (2.02 + 3.50i)10-s + (−2.42 − 4.20i)11-s + 0.999·12-s − 0.692·14-s + (2.02 + 3.50i)15-s + (−0.5 − 0.866i)16-s + (−3.69 + 6.39i)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 − 1.81·5-s + (−0.204 + 0.353i)6-s + (0.130 − 0.226i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.640 + 1.10i)10-s + (−0.731 − 1.26i)11-s + 0.288·12-s − 0.184·14-s + (0.522 + 0.905i)15-s + (−0.125 − 0.216i)16-s + (−0.895 + 1.55i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.997 - 0.0743i$
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: \((2,\ 1014,\ (\ :1/2),\ 0.997 - 0.0743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4056734388\)
\(L(\frac12)\) \(\approx\) \(0.4056734388\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.04T + 5T^{2} \)
7 \( 1 + (-0.346 + 0.599i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.42 + 4.20i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.69 - 6.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.890 - 1.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.55 + 4.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.67 - 2.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.972T + 31T^{2} \)
37 \( 1 + (-0.643 - 1.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.753 - 1.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.15 + 7.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.20T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + (-0.653 + 1.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.198 + 0.343i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.02 - 5.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.664 - 1.15i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.65T + 73T^{2} \)
79 \( 1 + 8.33T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (-1.55 - 2.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.27 - 7.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.54835537917556694638965515564, −8.642995582882031331478091297310, −8.455282940403414151994011505621, −7.69477096072419287580573089217, −6.82378217365940662220306316965, −5.68870312672567286846009845755, −4.31006936955463626061151295371, −3.73817959603087793365891493620, −2.52886471278543985345696762403, −0.815382014382314605043166285658, 0.32242968071794318824857439158, 2.61321101798368069192324880189, 4.08723601219882637699709604046, 4.62644936609091182875544150714, 5.49643995052538189355487938708, 6.99276964545003611898767648036, 7.35085461149932380982505756108, 8.181772681188123113590626667462, 9.039724008567746920369068107234, 9.830619882369048892991652409020

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