↑

Properties

Label 2-2217-2217.1115-c0-0-0
Degree $2$
Conductor $2217$
Sign $-0.0976 - 0.995i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.606 + 0.795i)3-s + (0.953 + 0.301i)4-s + (−1.97 − 0.304i)7-s + (−0.264 + 0.964i)9-s + (0.338 + 0.941i)12-s + (0.885 + 0.993i)13-s + (0.817 + 0.575i)16-s + (0.0551 + 1.43i)19-s + (−0.952 − 1.75i)21-s + (−0.264 − 0.964i)25-s + (−0.927 + 0.373i)27-s + (−1.78 − 0.884i)28-s + (0.352 + 0.395i)31-s + (−0.543 + 0.839i)36-s + (0.311 + 0.219i)37-s + ⋯
L(s)  = 1  + (0.606 + 0.795i)3-s + (0.953 + 0.301i)4-s + (−1.97 − 0.304i)7-s + (−0.264 + 0.964i)9-s + (0.338 + 0.941i)12-s + (0.885 + 0.993i)13-s + (0.817 + 0.575i)16-s + (0.0551 + 1.43i)19-s + (−0.952 − 1.75i)21-s + (−0.264 − 0.964i)25-s + (−0.927 + 0.373i)27-s + (−1.78 − 0.884i)28-s + (0.352 + 0.395i)31-s + (−0.543 + 0.839i)36-s + (0.311 + 0.219i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0976 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0976 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $-0.0976 - 0.995i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ -0.0976 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436925299\)
\(L(\frac12)\) \(\approx\) \(1.436925299\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.606 - 0.795i)T \)
739 \( 1 + (-0.338 - 0.941i)T \)
good2 \( 1 + (-0.953 - 0.301i)T^{2} \)
5 \( 1 + (0.264 + 0.964i)T^{2} \)
7 \( 1 + (1.97 + 0.304i)T + (0.953 + 0.301i)T^{2} \)
11 \( 1 + (0.973 - 0.227i)T^{2} \)
13 \( 1 + (-0.885 - 0.993i)T + (-0.114 + 0.993i)T^{2} \)
17 \( 1 + (-0.988 - 0.152i)T^{2} \)
19 \( 1 + (-0.0551 - 1.43i)T + (-0.997 + 0.0765i)T^{2} \)
23 \( 1 + (-0.953 - 0.301i)T^{2} \)
29 \( 1 + (0.409 + 0.912i)T^{2} \)
31 \( 1 + (-0.352 - 0.395i)T + (-0.114 + 0.993i)T^{2} \)
37 \( 1 + (-0.311 - 0.219i)T + (0.338 + 0.941i)T^{2} \)
41 \( 1 + (0.665 - 0.746i)T^{2} \)
43 \( 1 + (1.33 + 0.538i)T + (0.720 + 0.693i)T^{2} \)
47 \( 1 + (-0.896 - 0.443i)T^{2} \)
53 \( 1 + (-0.338 - 0.941i)T^{2} \)
59 \( 1 + (-0.720 + 0.693i)T^{2} \)
61 \( 1 + (-0.781 + 1.43i)T + (-0.543 - 0.839i)T^{2} \)
67 \( 1 + (0.505 - 1.83i)T + (-0.859 - 0.511i)T^{2} \)
71 \( 1 + (-0.896 + 0.443i)T^{2} \)
73 \( 1 + (-0.455 + 0.270i)T + (0.477 - 0.878i)T^{2} \)
79 \( 1 + (-0.376 + 1.94i)T + (-0.927 - 0.373i)T^{2} \)
83 \( 1 + (0.927 + 0.373i)T^{2} \)
89 \( 1 + (0.114 + 0.993i)T^{2} \)
97 \( 1 + (-0.668 - 0.103i)T + (0.953 + 0.301i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.573825223081236857551726369771, −8.690823144386714764717410713252, −8.006882911505332613313651372124, −6.98124956292287231966776854945, −6.39778721296277519305479921086, −5.74499896670299892940071448382, −4.20909713506660595675421945426, −3.57540961903509582003704538934, −3.00425810998272477927804197041, −1.88676744761026350162059081309, 0.899452689119469130342317470891, 2.36741204143568158025816095824, 3.05611966621095745773476144414, 3.60928978147958332286558562476, 5.44732629013661038234462698311, 6.18414204414871683414758806628, 6.66923551616254523549798202248, 7.29888728919415999746006960683, 8.182977762142915311658298462681, 9.116422822715409093184180604692

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