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Properties

Label 2-2217-2217.1748-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.756 - 0.653i$
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  + (0.0383 − 0.999i)3-s + (−0.859 − 0.511i)4-s + (−0.321 + 1.16i)7-s + (−0.997 − 0.0765i)9-s + (−0.543 + 0.839i)12-s + (1.19 + 0.981i)13-s + (0.477 + 0.878i)16-s + (−1.74 + 0.864i)19-s + (1.15 + 0.365i)21-s + (−0.997 + 0.0765i)25-s + (−0.114 + 0.993i)27-s + (0.873 − 0.840i)28-s + (1.53 + 1.26i)31-s + (0.817 + 0.575i)36-s + (−0.635 − 1.16i)37-s + ⋯
L(s)  = 1  + (0.0383 − 0.999i)3-s + (−0.859 − 0.511i)4-s + (−0.321 + 1.16i)7-s + (−0.997 − 0.0765i)9-s + (−0.543 + 0.839i)12-s + (1.19 + 0.981i)13-s + (0.477 + 0.878i)16-s + (−1.74 + 0.864i)19-s + (1.15 + 0.365i)21-s + (−0.997 + 0.0765i)25-s + (−0.114 + 0.993i)27-s + (0.873 − 0.840i)28-s + (1.53 + 1.26i)31-s + (0.817 + 0.575i)36-s + (−0.635 − 1.16i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.756 - 0.653i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.756 - 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6926531109\)
\(L(\frac12)\) \(\approx\) \(0.6926531109\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0383 + 0.999i)T \)
739 \( 1 + (0.543 - 0.839i)T \)
good2 \( 1 + (0.859 + 0.511i)T^{2} \)
5 \( 1 + (0.997 - 0.0765i)T^{2} \)
7 \( 1 + (0.321 - 1.16i)T + (-0.859 - 0.511i)T^{2} \)
11 \( 1 + (0.927 + 0.373i)T^{2} \)
13 \( 1 + (-1.19 - 0.981i)T + (0.190 + 0.981i)T^{2} \)
17 \( 1 + (0.264 - 0.964i)T^{2} \)
19 \( 1 + (1.74 - 0.864i)T + (0.606 - 0.795i)T^{2} \)
23 \( 1 + (0.859 + 0.511i)T^{2} \)
29 \( 1 + (-0.338 - 0.941i)T^{2} \)
31 \( 1 + (-1.53 - 1.26i)T + (0.190 + 0.981i)T^{2} \)
37 \( 1 + (0.635 + 1.16i)T + (-0.543 + 0.839i)T^{2} \)
41 \( 1 + (0.771 - 0.636i)T^{2} \)
43 \( 1 + (-0.223 - 1.93i)T + (-0.973 + 0.227i)T^{2} \)
47 \( 1 + (-0.720 + 0.693i)T^{2} \)
53 \( 1 + (0.543 - 0.839i)T^{2} \)
59 \( 1 + (0.973 + 0.227i)T^{2} \)
61 \( 1 + (-0.910 + 0.288i)T + (0.817 - 0.575i)T^{2} \)
67 \( 1 + (-1.71 - 0.131i)T + (0.988 + 0.152i)T^{2} \)
71 \( 1 + (-0.720 - 0.693i)T^{2} \)
73 \( 1 + (1.97 - 0.304i)T + (0.953 - 0.301i)T^{2} \)
79 \( 1 + (-0.352 + 0.395i)T + (-0.114 - 0.993i)T^{2} \)
83 \( 1 + (0.114 + 0.993i)T^{2} \)
89 \( 1 + (-0.190 + 0.981i)T^{2} \)
97 \( 1 + (-0.288 + 1.04i)T + (-0.859 - 0.511i)T^{2} \)
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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

−9.045627556857483413096544286288, −8.561195152911733581123137286561, −8.111921061083651866490578357498, −6.72468708925463851017212873361, −6.12730461778168889825896892331, −5.70715395160752610074651317969, −4.52933006828173136572009711209, −3.58242337681215974090198956707, −2.28727373773349316222974385589, −1.42348191998838598089379011580, 0.51341430764069763606594118311, 2.68240184790967761020480210350, 3.81649774496122171646653524197, 4.00569198151483441777690997890, 4.92842016375047656863872516250, 5.89598659884649842435148756071, 6.77384510942699594807919583007, 7.968159415368362673555161218222, 8.404004499681756678330670342287, 9.090979385028644486587922420338

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