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Properties

Label 2-1014-13.3-c1-0-9
Degree $2$
Conductor $1014$
Sign $0.314 - 0.949i$
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 − 0.267·5-s + (0.499 − 0.866i)6-s + (0.366 − 0.633i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.133 − 0.232i)10-s + (2.36 + 4.09i)11-s + 0.999·12-s + 0.732·14-s + (0.133 + 0.232i)15-s + (−0.5 − 0.866i)16-s + (1.13 − 1.96i)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 − 0.119·5-s + (0.204 − 0.353i)6-s + (0.138 − 0.239i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0423 − 0.0733i)10-s + (0.713 + 1.23i)11-s + 0.288·12-s + 0.195·14-s + (0.0345 + 0.0599i)15-s + (−0.125 − 0.216i)16-s + (0.275 − 0.476i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.314 - 0.949i$
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: \((2,\ 1014,\ (\ :1/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.644593410\)
\(L(\frac12)\) \(\approx\) \(1.644593410\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.267T + 5T^{2} \)
7 \( 1 + (-0.366 + 0.633i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.13 + 1.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.633 + 1.09i)T + (-9.5 - 16.4i)T^{2} \)
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 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + (-5.23 - 9.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.69 - 9.86i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.83 - 6.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.19T + 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.598 - 1.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.56 + 9.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.633 + 1.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.73T + 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (1.26 + 2.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3 - 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.830105109246211745108392398809, −9.408478388068381273718138946163, −8.075356903388568073950809278973, −7.53031331776512806227024603517, −6.76226750627744034975794504657, −5.98963947299702177873416977959, −4.89472731840353758347188571777, −4.19876384096816739662899619372, −2.84527503616467055042524689961, −1.32568421983442431747852034371, 0.801631083450501054090742212136, 2.43066041189292479433514768450, 3.64595289559704178267974483901, 4.23871446153835812201102453637, 5.54892460857752470125586669361, 5.97133185816806069970240002975, 7.17835112967168082002973841602, 8.525906878756154184702213142044, 8.946300670714311128294856217236, 10.00552297153686464281745876050

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