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Properties

Label 2-1014-13.9-c1-0-11
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 + 3.73·5-s + (−0.499 − 0.866i)6-s + (1.36 + 2.36i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.86 + 3.23i)10-s + (−0.633 + 1.09i)11-s + 0.999·12-s − 2.73·14-s + (−1.86 + 3.23i)15-s + (−0.5 + 0.866i)16-s + (2.86 + 4.96i)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 + 1.66·5-s + (−0.204 − 0.353i)6-s + (0.516 + 0.894i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.590 + 1.02i)10-s + (−0.191 + 0.331i)11-s + 0.288·12-s − 0.730·14-s + (−0.481 + 0.834i)15-s + (−0.125 + 0.216i)16-s + (0.695 + 1.20i)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: $\chi_{1014} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.314 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.641871938\)
\(L(\frac12)\) \(\approx\) \(1.641871938\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.73T + 5T^{2} \)
7 \( 1 + (-1.36 - 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.633 - 1.09i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.86 - 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.36 + 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.09 - 3.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.23 + 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + (1.76 - 3.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.69 + 8.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.83 - 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.19T + 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.56 - 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.36 + 4.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.26T + 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 + 0.196T + 83T^{2} \)
89 \( 1 + (-4.73 + 8.19i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.10258298234398457865634786862, −9.332847697381587169536813083680, −8.763872962134815614524881739550, −7.81546186649155360409167950632, −6.53760685400302860304977496968, −5.84494241582686315342978736907, −5.38068082764858988781042809711, −4.37724996921088537345360347029, −2.60764351709029808494873694966, −1.57571721381686050679496760927, 0.950900944682986773641632937442, 1.92012898038350777096510378350, 2.95819985223216164617315160464, 4.46290468013006483651713456094, 5.45045609462457575957114398073, 6.26613876679924249381235977495, 7.24475069030152342471311995898, 8.088588779179650751188415835641, 9.059894377773972190069464777896, 9.894298628793298207221251997172

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