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Properties

Label 2-2217-2217.1016-c0-0-0
Degree $2$
Conductor $2217$
Sign $-0.422 + 0.906i$
Analytic cond. $1.10642$
Root an. cond. $1.05186$
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  + (0.817 − 0.575i)3-s + (−0.665 − 0.746i)4-s + (0.444 − 0.992i)7-s + (0.338 − 0.941i)9-s + (−0.973 − 0.227i)12-s + (0.00293 + 0.0765i)13-s + (−0.114 + 0.993i)16-s + (−0.821 − 1.51i)19-s + (−0.206 − 1.06i)21-s + (0.338 + 0.941i)25-s + (−0.264 − 0.964i)27-s + (−1.03 + 0.328i)28-s + (0.0258 + 0.675i)31-s + (−0.927 + 0.373i)36-s + (−0.139 + 1.20i)37-s + ⋯
L(s)  = 1  + (0.817 − 0.575i)3-s + (−0.665 − 0.746i)4-s + (0.444 − 0.992i)7-s + (0.338 − 0.941i)9-s + (−0.973 − 0.227i)12-s + (0.00293 + 0.0765i)13-s + (−0.114 + 0.993i)16-s + (−0.821 − 1.51i)19-s + (−0.206 − 1.06i)21-s + (0.338 + 0.941i)25-s + (−0.264 − 0.964i)27-s + (−1.03 + 0.328i)28-s + (0.0258 + 0.675i)31-s + (−0.927 + 0.373i)36-s + (−0.139 + 1.20i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $-0.422 + 0.906i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ -0.422 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324934588\)
\(L(\frac12)\) \(\approx\) \(1.324934588\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.817 + 0.575i)T \)
739 \( 1 + (0.973 + 0.227i)T \)
good2 \( 1 + (0.665 + 0.746i)T^{2} \)
5 \( 1 + (-0.338 - 0.941i)T^{2} \)
7 \( 1 + (-0.444 + 0.992i)T + (-0.665 - 0.746i)T^{2} \)
11 \( 1 + (-0.988 - 0.152i)T^{2} \)
13 \( 1 + (-0.00293 - 0.0765i)T + (-0.997 + 0.0765i)T^{2} \)
17 \( 1 + (0.409 - 0.912i)T^{2} \)
19 \( 1 + (0.821 + 1.51i)T + (-0.543 + 0.839i)T^{2} \)
23 \( 1 + (0.665 + 0.746i)T^{2} \)
29 \( 1 + (-0.720 + 0.693i)T^{2} \)
31 \( 1 + (-0.0258 - 0.675i)T + (-0.997 + 0.0765i)T^{2} \)
37 \( 1 + (0.139 - 1.20i)T + (-0.973 - 0.227i)T^{2} \)
41 \( 1 + (-0.0383 + 0.999i)T^{2} \)
43 \( 1 + (-0.455 + 1.65i)T + (-0.859 - 0.511i)T^{2} \)
47 \( 1 + (-0.953 + 0.301i)T^{2} \)
53 \( 1 + (0.973 + 0.227i)T^{2} \)
59 \( 1 + (0.859 - 0.511i)T^{2} \)
61 \( 1 + (0.0436 - 0.225i)T + (-0.927 - 0.373i)T^{2} \)
67 \( 1 + (0.449 - 1.25i)T + (-0.771 - 0.636i)T^{2} \)
71 \( 1 + (-0.953 - 0.301i)T^{2} \)
73 \( 1 + (0.521 - 0.430i)T + (0.190 - 0.981i)T^{2} \)
79 \( 1 + (0.495 + 0.650i)T + (-0.264 + 0.964i)T^{2} \)
83 \( 1 + (0.264 - 0.964i)T^{2} \)
89 \( 1 + (0.997 + 0.0765i)T^{2} \)
97 \( 1 + (-0.796 + 1.77i)T + (-0.665 - 0.746i)T^{2} \)
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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

−8.778634359927809265141995180109, −8.503266675705843368375212110935, −7.26795633635784395546486117098, −6.95239788487894747082282650895, −5.89341945824788260148763397363, −4.79129535297494005429925138557, −4.20598593686509249507791284854, −3.15211018563858538022916536594, −1.88661300598299222769701619950, −0.885355653302459709642895352904, 2.00511662198359344253540356637, 2.85016679389587052703569861878, 3.83868580964111976896957331771, 4.47241565664571635132861698350, 5.33622021037187991189135664765, 6.26378947569982207434105548833, 7.66784578351127009238884498715, 8.040769759963581970157249808481, 8.728751322811518214927177154138, 9.245039707398618569874877637749

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