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Properties

Label 2-2717-2717.2001-c0-0-6
Degree $2$
Conductor $2717$
Sign $-0.571 + 0.820i$
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  + (1.10 − 0.924i)2-s + (−0.823 − 0.299i)3-s + (0.185 − 1.05i)4-s + (−1.18 + 0.431i)6-s + (0.990 − 1.71i)7-s + (−0.0502 − 0.0869i)8-s + (−0.177 − 0.148i)9-s + (0.5 + 0.866i)11-s + (−0.468 + 0.812i)12-s + (0.939 − 0.342i)13-s + (−0.494 − 2.80i)14-s + (0.869 + 0.316i)16-s − 0.332·18-s + (−0.559 + 0.829i)19-s + (−1.33 + 1.11i)21-s + (1.35 + 0.492i)22-s + ⋯
L(s)  = 1  + (1.10 − 0.924i)2-s + (−0.823 − 0.299i)3-s + (0.185 − 1.05i)4-s + (−1.18 + 0.431i)6-s + (0.990 − 1.71i)7-s + (−0.0502 − 0.0869i)8-s + (−0.177 − 0.148i)9-s + (0.5 + 0.866i)11-s + (−0.468 + 0.812i)12-s + (0.939 − 0.342i)13-s + (−0.494 − 2.80i)14-s + (0.869 + 0.316i)16-s − 0.332·18-s + (−0.559 + 0.829i)19-s + (−1.33 + 1.11i)21-s + (1.35 + 0.492i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.571 + 0.820i$
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: \((2,\ 2717,\ (\ :0),\ -0.571 + 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.915668988\)
\(L(\frac12)\) \(\approx\) \(1.915668988\)
\(L(1)\) 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 \( 1 + (0.559 - 0.829i)T \)
good2 \( 1 + (-1.10 + 0.924i)T + (0.173 - 0.984i)T^{2} \)
3 \( 1 + (0.823 + 0.299i)T + (0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.990 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.294 + 1.67i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.704 + 0.256i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.346 - 1.96i)T + (-0.939 - 0.342i)T^{2} \)
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 \( 1 + (1.52 + 0.553i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.882 - 1.52i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.650636246835928693130425061953, −7.891148679671358430990789352701, −7.04684833024275400620425604501, −6.26377361515947968128016360461, −5.48179767848507466538978726614, −4.45784956493841977836589521571, −4.20914738370697755365772661712, −3.28234304198047603204384297695, −1.83677945416074582568965599177, −1.09263303279911739270703398239, 1.70042152663876782877143037590, 3.04035904165380091798070171343, 4.06056544538674418730759128078, 4.97851245899415795457430813039, 5.42900970329363063950860733613, 6.03112996828884271150662243977, 6.46334539248456435624051355766, 7.67289667776896416876260414592, 8.477931670551389289195040333848, 8.932404028084394284721310903403

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