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Properties

Label 2-2717-2717.2001-c0-0-2
Degree $2$
Conductor $2717$
Sign $0.603 + 0.797i$
Analytic cond. $1.35595$
Root an. cond. $1.16445$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 − 1.13i)2-s + (−1.86 − 0.677i)3-s + (0.367 − 2.08i)4-s + (−3.28 + 1.19i)6-s + (−0.719 + 1.24i)7-s + (−0.987 − 1.71i)8-s + (2.23 + 1.87i)9-s + (0.5 + 0.866i)11-s + (−2.09 + 3.63i)12-s + (0.939 − 0.342i)13-s + (0.441 + 2.50i)14-s + (−1.28 − 0.468i)16-s + 5.16·18-s + (0.615 + 0.788i)19-s + (2.18 − 1.83i)21-s + (1.65 + 0.603i)22-s + ⋯
L(s)  = 1  + (1.35 − 1.13i)2-s + (−1.86 − 0.677i)3-s + (0.367 − 2.08i)4-s + (−3.28 + 1.19i)6-s + (−0.719 + 1.24i)7-s + (−0.987 − 1.71i)8-s + (2.23 + 1.87i)9-s + (0.5 + 0.866i)11-s + (−2.09 + 3.63i)12-s + (0.939 − 0.342i)13-s + (0.441 + 2.50i)14-s + (−1.28 − 0.468i)16-s + 5.16·18-s + (0.615 + 0.788i)19-s + (2.18 − 1.83i)21-s + (1.65 + 0.603i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (2001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ 0.603 + 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295876357\)
\(L(\frac12)\) \(\approx\) \(1.295876357\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.615 - 0.788i)T \)
good2 \( 1 + (-1.35 + 1.13i)T + (0.173 - 0.984i)T^{2} \)
3 \( 1 + (1.86 + 0.677i)T + (0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.719 - 1.24i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.346 - 1.96i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.454 + 0.165i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.130 - 0.737i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.580 - 0.211i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.438 + 0.759i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T^{2} \)
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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

−9.484820676128452592964098639349, −7.86148483559458631606592568675, −6.92016303887971069709140455955, −6.01453324195178497880799551865, −5.75329655200092180740581848534, −5.22696502149361391090182721559, −4.19648953414862425483425817962, −3.35944607535613157255147338568, −1.96724413648700352812289084797, −1.40462800934705747054331333373, 0.74762704529894936654153735946, 3.39160415883714317957118840310, 3.99823293667939082587734706051, 4.48114440322735074444760190725, 5.34103706015611914228592190898, 6.11538097015229498595614891105, 6.60397266288087223140161576602, 6.86250015270898700346805591308, 7.993742600874135373443955999843, 9.147778531546102548164613038512

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