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Properties

Label 2-2217-2217.890-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.975 + 0.219i$
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  + (−0.543 − 0.839i)3-s + (0.190 + 0.981i)4-s + (−0.521 − 0.430i)7-s + (−0.409 + 0.912i)9-s + (0.720 − 0.693i)12-s + (0.735 − 0.964i)13-s + (−0.927 + 0.373i)16-s + (1.55 + 1.09i)19-s + (−0.0775 + 0.671i)21-s + (−0.409 − 0.912i)25-s + (0.988 − 0.152i)27-s + (0.322 − 0.593i)28-s + (−0.495 + 0.650i)31-s + (−0.973 − 0.227i)36-s + (1.84 − 0.745i)37-s + ⋯
L(s)  = 1  + (−0.543 − 0.839i)3-s + (0.190 + 0.981i)4-s + (−0.521 − 0.430i)7-s + (−0.409 + 0.912i)9-s + (0.720 − 0.693i)12-s + (0.735 − 0.964i)13-s + (−0.927 + 0.373i)16-s + (1.55 + 1.09i)19-s + (−0.0775 + 0.671i)21-s + (−0.409 − 0.912i)25-s + (0.988 − 0.152i)27-s + (0.322 − 0.593i)28-s + (−0.495 + 0.650i)31-s + (−0.973 − 0.227i)36-s + (1.84 − 0.745i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.975 + 0.219i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (890, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.975 + 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9950079964\)
\(L(\frac12)\) \(\approx\) \(0.9950079964\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.543 + 0.839i)T \)
739 \( 1 + (-0.720 + 0.693i)T \)
good2 \( 1 + (-0.190 - 0.981i)T^{2} \)
5 \( 1 + (0.409 + 0.912i)T^{2} \)
7 \( 1 + (0.521 + 0.430i)T + (0.190 + 0.981i)T^{2} \)
11 \( 1 + (0.859 + 0.511i)T^{2} \)
13 \( 1 + (-0.735 + 0.964i)T + (-0.264 - 0.964i)T^{2} \)
17 \( 1 + (0.771 + 0.636i)T^{2} \)
19 \( 1 + (-1.55 - 1.09i)T + (0.338 + 0.941i)T^{2} \)
23 \( 1 + (-0.190 - 0.981i)T^{2} \)
29 \( 1 + (-0.896 - 0.443i)T^{2} \)
31 \( 1 + (0.495 - 0.650i)T + (-0.264 - 0.964i)T^{2} \)
37 \( 1 + (-1.84 + 0.745i)T + (0.720 - 0.693i)T^{2} \)
41 \( 1 + (-0.606 - 0.795i)T^{2} \)
43 \( 1 + (-1.88 - 0.291i)T + (0.953 + 0.301i)T^{2} \)
47 \( 1 + (-0.477 + 0.878i)T^{2} \)
53 \( 1 + (-0.720 + 0.693i)T^{2} \)
59 \( 1 + (-0.953 + 0.301i)T^{2} \)
61 \( 1 + (-0.212 - 1.84i)T + (-0.973 + 0.227i)T^{2} \)
67 \( 1 + (0.155 - 0.347i)T + (-0.665 - 0.746i)T^{2} \)
71 \( 1 + (-0.477 - 0.878i)T^{2} \)
73 \( 1 + (-0.544 + 0.610i)T + (-0.114 - 0.993i)T^{2} \)
79 \( 1 + (-1.53 - 0.118i)T + (0.988 + 0.152i)T^{2} \)
83 \( 1 + (-0.988 - 0.152i)T^{2} \)
89 \( 1 + (0.264 - 0.964i)T^{2} \)
97 \( 1 + (1.11 + 0.916i)T + (0.190 + 0.981i)T^{2} \)
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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

−9.072231170559879050845081040052, −8.033920809496463109923542569377, −7.73523007368921665810912462363, −7.00582557510302857886094615595, −6.09652020072379645439799605360, −5.54223798275926402870788839569, −4.20986719249606851670517495346, −3.35388207631444030985940210930, −2.47331400693236832967590446456, −1.01645481105212322227627648999, 1.03287101958348078237096765390, 2.52433305983205624925975462318, 3.61772574861704351596008458560, 4.59250441449335773976361957150, 5.36615113548225880544001124402, 6.04419932481009586469944911162, 6.61305183986005338737378440291, 7.59040659713507538910227567591, 8.985168283975075135267202278136, 9.486859819362780090380045952591

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