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Properties

Label 2-2717-2717.1429-c0-0-1
Degree $2$
Conductor $2717$
Sign $-0.999 - 0.00406i$
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  + (−0.704 − 0.256i)2-s + (0.294 + 1.67i)3-s + (−0.336 − 0.281i)4-s + (0.220 − 1.25i)6-s + (−0.997 + 1.72i)7-s + (0.538 + 0.933i)8-s + (−1.76 + 0.641i)9-s + (0.5 + 0.866i)11-s + (0.372 − 0.644i)12-s + (−0.173 + 0.984i)13-s + (1.14 − 0.960i)14-s + (−0.0640 − 0.363i)16-s + 1.40·18-s + (0.882 + 0.469i)19-s + (−3.17 − 1.15i)21-s + (−0.130 − 0.737i)22-s + ⋯
L(s)  = 1  + (−0.704 − 0.256i)2-s + (0.294 + 1.67i)3-s + (−0.336 − 0.281i)4-s + (0.220 − 1.25i)6-s + (−0.997 + 1.72i)7-s + (0.538 + 0.933i)8-s + (−1.76 + 0.641i)9-s + (0.5 + 0.866i)11-s + (0.372 − 0.644i)12-s + (−0.173 + 0.984i)13-s + (1.14 − 0.960i)14-s + (−0.0640 − 0.363i)16-s + 1.40·18-s + (0.882 + 0.469i)19-s + (−3.17 − 1.15i)21-s + (−0.130 − 0.737i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.999 - 0.00406i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ -0.999 - 0.00406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6925744514\)
\(L(\frac12)\) \(\approx\) \(0.6925744514\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-0.882 - 0.469i)T \)
good2 \( 1 + (0.704 + 0.256i)T + (0.766 + 0.642i)T^{2} \)
3 \( 1 + (-0.294 - 1.67i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.997 - 1.72i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-1.47 - 1.23i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.194 + 1.10i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.0534 - 0.0448i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.107 + 0.608i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.241 - 0.419i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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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

−9.492911259230214549583903062635, −8.970122167638909142518201231783, −8.532032462961314382934868770412, −7.24139687151277443545736317789, −6.09956775014749025130984315609, −5.29826775245983227944692453889, −4.79587431981827564506954454038, −3.78561910821738734533704110148, −2.89392507396070023866349931144, −1.92847961337346458341436017613, 0.69120150386144275968211190354, 1.08227980969358040978710317496, 3.06548581307168029416230445041, 3.36460270635726673860180272119, 4.66780534965741679341029785540, 6.01857622370776221329259858027, 6.88156721066162718496196831450, 7.11852452913230438063685442418, 7.75881881663070393191640549874, 8.460908566100001911613754732679

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