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Properties

Label 2-2217-2217.1139-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.899 - 0.436i$
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  + (0.896 + 0.443i)3-s + (0.988 + 0.152i)4-s + (−0.0763 − 0.00586i)7-s + (0.606 + 0.795i)9-s + (0.817 + 0.575i)12-s + (0.334 − 0.746i)13-s + (0.953 + 0.301i)16-s + (−1.33 − 1.28i)19-s + (−0.0658 − 0.0391i)21-s + (0.606 − 0.795i)25-s + (0.190 + 0.981i)27-s + (−0.0745 − 0.0174i)28-s + (−0.495 + 1.10i)31-s + (0.477 + 0.878i)36-s + (−1.47 − 0.465i)37-s + ⋯
L(s)  = 1  + (0.896 + 0.443i)3-s + (0.988 + 0.152i)4-s + (−0.0763 − 0.00586i)7-s + (0.606 + 0.795i)9-s + (0.817 + 0.575i)12-s + (0.334 − 0.746i)13-s + (0.953 + 0.301i)16-s + (−1.33 − 1.28i)19-s + (−0.0658 − 0.0391i)21-s + (0.606 − 0.795i)25-s + (0.190 + 0.981i)27-s + (−0.0745 − 0.0174i)28-s + (−0.495 + 1.10i)31-s + (0.477 + 0.878i)36-s + (−1.47 − 0.465i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.899 - 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.997560984\)
\(L(\frac12)\) \(\approx\) \(1.997560984\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.896 - 0.443i)T \)
739 \( 1 + (-0.817 - 0.575i)T \)
good2 \( 1 + (-0.988 - 0.152i)T^{2} \)
5 \( 1 + (-0.606 + 0.795i)T^{2} \)
7 \( 1 + (0.0763 + 0.00586i)T + (0.988 + 0.152i)T^{2} \)
11 \( 1 + (0.114 + 0.993i)T^{2} \)
13 \( 1 + (-0.334 + 0.746i)T + (-0.665 - 0.746i)T^{2} \)
17 \( 1 + (0.997 + 0.0765i)T^{2} \)
19 \( 1 + (1.33 + 1.28i)T + (0.0383 + 0.999i)T^{2} \)
23 \( 1 + (-0.988 - 0.152i)T^{2} \)
29 \( 1 + (0.543 - 0.839i)T^{2} \)
31 \( 1 + (0.495 - 1.10i)T + (-0.665 - 0.746i)T^{2} \)
37 \( 1 + (1.47 + 0.465i)T + (0.817 + 0.575i)T^{2} \)
41 \( 1 + (0.409 + 0.912i)T^{2} \)
43 \( 1 + (0.353 - 1.82i)T + (-0.927 - 0.373i)T^{2} \)
47 \( 1 + (0.973 + 0.227i)T^{2} \)
53 \( 1 + (-0.817 - 0.575i)T^{2} \)
59 \( 1 + (0.927 - 0.373i)T^{2} \)
61 \( 1 + (1.63 - 0.974i)T + (0.477 - 0.878i)T^{2} \)
67 \( 1 + (-1.19 - 1.57i)T + (-0.264 + 0.964i)T^{2} \)
71 \( 1 + (0.973 - 0.227i)T^{2} \)
73 \( 1 + (0.321 + 1.16i)T + (-0.859 + 0.511i)T^{2} \)
79 \( 1 + (-1.53 + 1.26i)T + (0.190 - 0.981i)T^{2} \)
83 \( 1 + (-0.190 + 0.981i)T^{2} \)
89 \( 1 + (0.665 - 0.746i)T^{2} \)
97 \( 1 + (1.63 + 0.125i)T + (0.988 + 0.152i)T^{2} \)
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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

−9.120811731628333600621315729805, −8.507814422477043512200929426945, −7.86328205617074289866706844065, −6.97049304004520439249270363808, −6.40084950732044734175623848175, −5.23345907068892130006078163886, −4.34568925718265540538592765491, −3.25833168188182592321856094267, −2.70801674491714232824102743105, −1.65748064995160682919811025553, 1.60602668571912997763613069236, 2.14993042147234772868912333080, 3.34443163369885600274536029376, 3.98085952157533367448470195445, 5.35770633263075437246824197435, 6.43330908728351923656481895223, 6.74598320634783025853239591532, 7.68234495024420926623805118783, 8.299838407931199803730596714039, 9.087869082972480574596184405178

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