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Properties

Label 2-2217-2217.191-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.895 - 0.445i$
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  + (0.720 − 0.693i)3-s + (−0.264 + 0.964i)4-s + (1.08 + 1.42i)7-s + (0.0383 − 0.999i)9-s + (0.477 + 0.878i)12-s + (0.228 − 0.636i)13-s + (−0.859 − 0.511i)16-s + (0.223 − 0.0522i)19-s + (1.77 + 0.273i)21-s + (0.0383 + 0.999i)25-s + (−0.665 − 0.746i)27-s + (−1.66 + 0.670i)28-s + (0.0258 − 0.0720i)31-s + (0.953 + 0.301i)36-s + (0.703 + 0.418i)37-s + ⋯
L(s)  = 1  + (0.720 − 0.693i)3-s + (−0.264 + 0.964i)4-s + (1.08 + 1.42i)7-s + (0.0383 − 0.999i)9-s + (0.477 + 0.878i)12-s + (0.228 − 0.636i)13-s + (−0.859 − 0.511i)16-s + (0.223 − 0.0522i)19-s + (1.77 + 0.273i)21-s + (0.0383 + 0.999i)25-s + (−0.665 − 0.746i)27-s + (−1.66 + 0.670i)28-s + (0.0258 − 0.0720i)31-s + (0.953 + 0.301i)36-s + (0.703 + 0.418i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.895 - 0.445i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.895 - 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.588976899\)
\(L(\frac12)\) \(\approx\) \(1.588976899\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.720 + 0.693i)T \)
739 \( 1 + (-0.477 - 0.878i)T \)
good2 \( 1 + (0.264 - 0.964i)T^{2} \)
5 \( 1 + (-0.0383 - 0.999i)T^{2} \)
7 \( 1 + (-1.08 - 1.42i)T + (-0.264 + 0.964i)T^{2} \)
11 \( 1 + (-0.190 + 0.981i)T^{2} \)
13 \( 1 + (-0.228 + 0.636i)T + (-0.771 - 0.636i)T^{2} \)
17 \( 1 + (-0.606 - 0.795i)T^{2} \)
19 \( 1 + (-0.223 + 0.0522i)T + (0.896 - 0.443i)T^{2} \)
23 \( 1 + (0.264 - 0.964i)T^{2} \)
29 \( 1 + (-0.817 - 0.575i)T^{2} \)
31 \( 1 + (-0.0258 + 0.0720i)T + (-0.771 - 0.636i)T^{2} \)
37 \( 1 + (-0.703 - 0.418i)T + (0.477 + 0.878i)T^{2} \)
41 \( 1 + (-0.338 - 0.941i)T^{2} \)
43 \( 1 + (-0.152 + 0.171i)T + (-0.114 - 0.993i)T^{2} \)
47 \( 1 + (0.927 - 0.373i)T^{2} \)
53 \( 1 + (-0.477 - 0.878i)T^{2} \)
59 \( 1 + (0.114 - 0.993i)T^{2} \)
61 \( 1 + (1.69 - 0.262i)T + (0.953 - 0.301i)T^{2} \)
67 \( 1 + (0.0202 - 0.529i)T + (-0.997 - 0.0765i)T^{2} \)
71 \( 1 + (0.927 + 0.373i)T^{2} \)
73 \( 1 + (0.0763 - 0.00586i)T + (0.988 - 0.152i)T^{2} \)
79 \( 1 + (0.495 + 1.10i)T + (-0.665 + 0.746i)T^{2} \)
83 \( 1 + (0.665 - 0.746i)T^{2} \)
89 \( 1 + (0.771 - 0.636i)T^{2} \)
97 \( 1 + (-0.579 - 0.759i)T + (-0.264 + 0.964i)T^{2} \)
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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

−8.933738766651073272117440512757, −8.518175064650881905723016547768, −7.83051915644629308300002197275, −7.34655521800140783206597341238, −6.18150987377048423962688984960, −5.35023925982788111357513401571, −4.38745868761986728654325988417, −3.23825511442889515483501383125, −2.61772875700632038667345207186, −1.58767083078161419581622229276, 1.20886773284824830237527836451, 2.21638555094569454425299211097, 3.67670822086115909894039462982, 4.47871752331198473334234121113, 4.80878177025269112518915273298, 5.93719149251138939206959242048, 6.96756748089951940820299244580, 7.77474638171073965583694407713, 8.453353060508164438310138975784, 9.286933288604329214496801578496

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