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Properties

Label 2-2217-2217.1349-c0-0-0
Degree $2$
Conductor $2217$
Sign $-0.200 + 0.979i$
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  + (−0.771 − 0.636i)3-s + (−0.973 − 0.227i)4-s + (0.152 − 1.32i)7-s + (0.190 + 0.981i)9-s + (0.606 + 0.795i)12-s + (1.95 − 0.301i)13-s + (0.896 + 0.443i)16-s + (0.444 + 0.992i)19-s + (−0.958 + 0.922i)21-s + (0.190 − 0.981i)25-s + (0.477 − 0.878i)27-s + (−0.449 + 1.25i)28-s + (0.376 − 0.0581i)31-s + (0.0383 − 0.999i)36-s + (−1.54 − 0.762i)37-s + ⋯
L(s)  = 1  + (−0.771 − 0.636i)3-s + (−0.973 − 0.227i)4-s + (0.152 − 1.32i)7-s + (0.190 + 0.981i)9-s + (0.606 + 0.795i)12-s + (1.95 − 0.301i)13-s + (0.896 + 0.443i)16-s + (0.444 + 0.992i)19-s + (−0.958 + 0.922i)21-s + (0.190 − 0.981i)25-s + (0.477 − 0.878i)27-s + (−0.449 + 1.25i)28-s + (0.376 − 0.0581i)31-s + (0.0383 − 0.999i)36-s + (−1.54 − 0.762i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $-0.200 + 0.979i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ -0.200 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7626586501\)
\(L(\frac12)\) \(\approx\) \(0.7626586501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.771 + 0.636i)T \)
739 \( 1 + (-0.606 - 0.795i)T \)
good2 \( 1 + (0.973 + 0.227i)T^{2} \)
5 \( 1 + (-0.190 + 0.981i)T^{2} \)
7 \( 1 + (-0.152 + 1.32i)T + (-0.973 - 0.227i)T^{2} \)
11 \( 1 + (-0.817 - 0.575i)T^{2} \)
13 \( 1 + (-1.95 + 0.301i)T + (0.953 - 0.301i)T^{2} \)
17 \( 1 + (0.114 - 0.993i)T^{2} \)
19 \( 1 + (-0.444 - 0.992i)T + (-0.665 + 0.746i)T^{2} \)
23 \( 1 + (0.973 + 0.227i)T^{2} \)
29 \( 1 + (0.997 + 0.0765i)T^{2} \)
31 \( 1 + (-0.376 + 0.0581i)T + (0.953 - 0.301i)T^{2} \)
37 \( 1 + (1.54 + 0.762i)T + (0.606 + 0.795i)T^{2} \)
41 \( 1 + (-0.988 - 0.152i)T^{2} \)
43 \( 1 + (0.519 + 0.955i)T + (-0.543 + 0.839i)T^{2} \)
47 \( 1 + (-0.338 + 0.941i)T^{2} \)
53 \( 1 + (-0.606 - 0.795i)T^{2} \)
59 \( 1 + (0.543 + 0.839i)T^{2} \)
61 \( 1 + (-1.29 - 1.24i)T + (0.0383 + 0.999i)T^{2} \)
67 \( 1 + (0.370 + 1.91i)T + (-0.927 + 0.373i)T^{2} \)
71 \( 1 + (-0.338 - 0.941i)T^{2} \)
73 \( 1 + (0.353 + 0.142i)T + (0.720 + 0.693i)T^{2} \)
79 \( 1 + (-0.197 - 0.117i)T + (0.477 + 0.878i)T^{2} \)
83 \( 1 + (-0.477 - 0.878i)T^{2} \)
89 \( 1 + (-0.953 - 0.301i)T^{2} \)
97 \( 1 + (0.139 - 1.20i)T + (-0.973 - 0.227i)T^{2} \)
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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

−8.804955061607755139315990145717, −8.227567217640412651915417894698, −7.50899796147022881997288268352, −6.58146771637361854137515971283, −5.87792134825297697478951216668, −5.12396646090099206321995171405, −4.12452995361914297999671301388, −3.56096690720688594783901645071, −1.58660410517588503506553599716, −0.74902783386661155442079700111, 1.29554509322675672283116892902, 3.08427399456480468829043219369, 3.80473967382542832785134596621, 4.82186321874570265796350290142, 5.38741527958138680619348372852, 6.07902300990557722083032284328, 6.94559789366956065228457970580, 8.351234909096999777908264403541, 8.756248212063127012904047832528, 9.315652733759164029151710363928

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