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Properties

Label 2-2217-2217.1277-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.488 - 0.872i$
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  + (−0.859 − 0.511i)3-s + (0.338 + 0.941i)4-s + (1.55 + 1.09i)7-s + (0.477 + 0.878i)9-s + (0.190 − 0.981i)12-s + (1.89 + 0.443i)13-s + (−0.771 + 0.636i)16-s + (−1.97 + 0.304i)19-s + (−0.780 − 1.73i)21-s + (0.477 − 0.878i)25-s + (0.0383 − 0.999i)27-s + (−0.505 + 1.83i)28-s + (−0.930 − 0.217i)31-s + (−0.665 + 0.746i)36-s + (−1.11 + 0.916i)37-s + ⋯
L(s)  = 1  + (−0.859 − 0.511i)3-s + (0.338 + 0.941i)4-s + (1.55 + 1.09i)7-s + (0.477 + 0.878i)9-s + (0.190 − 0.981i)12-s + (1.89 + 0.443i)13-s + (−0.771 + 0.636i)16-s + (−1.97 + 0.304i)19-s + (−0.780 − 1.73i)21-s + (0.477 − 0.878i)25-s + (0.0383 − 0.999i)27-s + (−0.505 + 1.83i)28-s + (−0.930 − 0.217i)31-s + (−0.665 + 0.746i)36-s + (−1.11 + 0.916i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175327548\)
\(L(\frac12)\) \(\approx\) \(1.175327548\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.859 + 0.511i)T \)
739 \( 1 + (-0.190 + 0.981i)T \)
good2 \( 1 + (-0.338 - 0.941i)T^{2} \)
5 \( 1 + (-0.477 + 0.878i)T^{2} \)
7 \( 1 + (-1.55 - 1.09i)T + (0.338 + 0.941i)T^{2} \)
11 \( 1 + (-0.606 + 0.795i)T^{2} \)
13 \( 1 + (-1.89 - 0.443i)T + (0.896 + 0.443i)T^{2} \)
17 \( 1 + (-0.817 - 0.575i)T^{2} \)
19 \( 1 + (1.97 - 0.304i)T + (0.953 - 0.301i)T^{2} \)
23 \( 1 + (-0.338 - 0.941i)T^{2} \)
29 \( 1 + (0.114 + 0.993i)T^{2} \)
31 \( 1 + (0.930 + 0.217i)T + (0.896 + 0.443i)T^{2} \)
37 \( 1 + (1.11 - 0.916i)T + (0.190 - 0.981i)T^{2} \)
41 \( 1 + (0.973 - 0.227i)T^{2} \)
43 \( 1 + (0.0763 + 1.99i)T + (-0.997 + 0.0765i)T^{2} \)
47 \( 1 + (0.264 - 0.964i)T^{2} \)
53 \( 1 + (-0.190 + 0.981i)T^{2} \)
59 \( 1 + (0.997 + 0.0765i)T^{2} \)
61 \( 1 + (-0.631 + 1.40i)T + (-0.665 - 0.746i)T^{2} \)
67 \( 1 + (-0.322 - 0.593i)T + (-0.543 + 0.839i)T^{2} \)
71 \( 1 + (0.264 + 0.964i)T^{2} \)
73 \( 1 + (0.519 + 0.801i)T + (-0.409 + 0.912i)T^{2} \)
79 \( 1 + (-1.17 - 1.13i)T + (0.0383 + 0.999i)T^{2} \)
83 \( 1 + (-0.0383 - 0.999i)T^{2} \)
89 \( 1 + (-0.896 + 0.443i)T^{2} \)
97 \( 1 + (-0.311 - 0.219i)T + (0.338 + 0.941i)T^{2} \)
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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

−8.864089086858128707223007708164, −8.442812995143487062651895048744, −8.053247678092122525451012583805, −6.89918526497761978536027989988, −6.31143591061933395154311901164, −5.52322386899842113357057438766, −4.58238606406330630059375294839, −3.78677821575362283264415886976, −2.21112624584784648281236162992, −1.71160119956611113483863746108, 1.01202041612800414683032561542, 1.77355479815918697437262254113, 3.66435902917860921284365425580, 4.41612621122782030440841470149, 5.12412899328148321032562509530, 5.90448424305025051112206463608, 6.61055579629303260732157976512, 7.39269665013416619903032420522, 8.463874151558208677913042869387, 9.077469643556653508406344780316

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