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Properties

Label 2-2217-2217.1106-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.757 - 0.653i$
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  + (−0.927 + 0.373i)3-s + (0.606 + 0.795i)4-s + (−1.74 − 0.864i)7-s + (0.720 − 0.693i)9-s + (−0.859 − 0.511i)12-s + (1.33 − 0.941i)13-s + (−0.264 + 0.964i)16-s + (0.152 + 1.32i)19-s + (1.94 + 0.149i)21-s + (0.720 + 0.693i)25-s + (−0.409 + 0.912i)27-s + (−0.370 − 1.91i)28-s + (1.17 − 0.829i)31-s + (0.988 + 0.152i)36-s + (0.288 − 1.04i)37-s + ⋯
L(s)  = 1  + (−0.927 + 0.373i)3-s + (0.606 + 0.795i)4-s + (−1.74 − 0.864i)7-s + (0.720 − 0.693i)9-s + (−0.859 − 0.511i)12-s + (1.33 − 0.941i)13-s + (−0.264 + 0.964i)16-s + (0.152 + 1.32i)19-s + (1.94 + 0.149i)21-s + (0.720 + 0.693i)25-s + (−0.409 + 0.912i)27-s + (−0.370 − 1.91i)28-s + (1.17 − 0.829i)31-s + (0.988 + 0.152i)36-s + (0.288 − 1.04i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.757 - 0.653i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.757 - 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8734199262\)
\(L(\frac12)\) \(\approx\) \(0.8734199262\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.927 - 0.373i)T \)
739 \( 1 + (0.859 + 0.511i)T \)
good2 \( 1 + (-0.606 - 0.795i)T^{2} \)
5 \( 1 + (-0.720 - 0.693i)T^{2} \)
7 \( 1 + (1.74 + 0.864i)T + (0.606 + 0.795i)T^{2} \)
11 \( 1 + (0.771 - 0.636i)T^{2} \)
13 \( 1 + (-1.33 + 0.941i)T + (0.338 - 0.941i)T^{2} \)
17 \( 1 + (-0.896 - 0.443i)T^{2} \)
19 \( 1 + (-0.152 - 1.32i)T + (-0.973 + 0.227i)T^{2} \)
23 \( 1 + (-0.606 - 0.795i)T^{2} \)
29 \( 1 + (-0.953 - 0.301i)T^{2} \)
31 \( 1 + (-1.17 + 0.829i)T + (0.338 - 0.941i)T^{2} \)
37 \( 1 + (-0.288 + 1.04i)T + (-0.859 - 0.511i)T^{2} \)
41 \( 1 + (-0.817 - 0.575i)T^{2} \)
43 \( 1 + (-0.544 - 1.21i)T + (-0.665 + 0.746i)T^{2} \)
47 \( 1 + (-0.190 - 0.981i)T^{2} \)
53 \( 1 + (0.859 + 0.511i)T^{2} \)
59 \( 1 + (0.665 + 0.746i)T^{2} \)
61 \( 1 + (-0.528 + 0.0405i)T + (0.988 - 0.152i)T^{2} \)
67 \( 1 + (-0.873 + 0.840i)T + (0.0383 - 0.999i)T^{2} \)
71 \( 1 + (-0.190 + 0.981i)T^{2} \)
73 \( 1 + (-0.0551 - 1.43i)T + (-0.997 + 0.0765i)T^{2} \)
79 \( 1 + (0.974 - 1.50i)T + (-0.409 - 0.912i)T^{2} \)
83 \( 1 + (0.409 + 0.912i)T^{2} \)
89 \( 1 + (-0.338 - 0.941i)T^{2} \)
97 \( 1 + (1.54 + 0.762i)T + (0.606 + 0.795i)T^{2} \)
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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

−9.605801825467382883929410492795, −8.485294603802613714325867706403, −7.60195492782854541320170372816, −6.86094875126692359722258377943, −6.17740499991468249506341697757, −5.73520905719021974559946501392, −4.15966007670164712557210224066, −3.64831858748015562500371665340, −2.94428030255879046366601087522, −1.03159037325346836765995153481, 0.891008107862959497939072739737, 2.23477736600098310927545286690, 3.14745352062559757048827665456, 4.52040177188033638821091819930, 5.46774121298694832904273959178, 6.24592647511474221011962625948, 6.57809622470140233770424327994, 7.06961218787110503131945568902, 8.578670882055030011710413082840, 9.220890454551544657178925042111

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