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Properties

Label 2-3311-3311.2631-c0-0-0
Degree $2$
Conductor $3311$
Sign $-0.478 - 0.878i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
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  + (1.65 + 0.988i)2-s + (1.28 + 2.39i)4-s + (0.809 + 0.587i)7-s + (−0.148 + 3.30i)8-s + (−0.134 − 0.990i)9-s + (−0.858 + 0.512i)11-s + (0.757 + 1.77i)14-s + (−2.01 + 3.05i)16-s + (0.757 − 1.77i)18-s − 1.92·22-s + (−0.0808 + 0.0389i)23-s + (−0.753 − 0.657i)25-s + (−0.364 + 2.69i)28-s + (0.202 − 0.176i)29-s + (−3.38 + 1.62i)32-s + ⋯
L(s)  = 1  + (1.65 + 0.988i)2-s + (1.28 + 2.39i)4-s + (0.809 + 0.587i)7-s + (−0.148 + 3.30i)8-s + (−0.134 − 0.990i)9-s + (−0.858 + 0.512i)11-s + (0.757 + 1.77i)14-s + (−2.01 + 3.05i)16-s + (0.757 − 1.77i)18-s − 1.92·22-s + (−0.0808 + 0.0389i)23-s + (−0.753 − 0.657i)25-s + (−0.364 + 2.69i)28-s + (0.202 − 0.176i)29-s + (−3.38 + 1.62i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (2631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.478 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.332745029\)
\(L(\frac12)\) \(\approx\) \(3.332745029\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (0.691 - 0.722i)T \)
good2 \( 1 + (-1.65 - 0.988i)T + (0.473 + 0.880i)T^{2} \)
3 \( 1 + (0.134 + 0.990i)T^{2} \)
5 \( 1 + (0.753 + 0.657i)T^{2} \)
13 \( 1 + (-0.393 + 0.919i)T^{2} \)
17 \( 1 + (-0.393 - 0.919i)T^{2} \)
19 \( 1 + (-0.936 + 0.351i)T^{2} \)
23 \( 1 + (0.0808 - 0.0389i)T + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.202 + 0.176i)T + (0.134 - 0.990i)T^{2} \)
31 \( 1 + (-0.473 - 0.880i)T^{2} \)
37 \( 1 + (-1.17 + 1.61i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.691 + 0.722i)T^{2} \)
47 \( 1 + (-0.936 + 0.351i)T^{2} \)
53 \( 1 + (-1.75 + 0.657i)T + (0.753 - 0.657i)T^{2} \)
59 \( 1 + (0.995 - 0.0896i)T^{2} \)
61 \( 1 + (-0.473 + 0.880i)T^{2} \)
67 \( 1 + (1.54 + 0.744i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-1.88 - 0.255i)T + (0.963 + 0.266i)T^{2} \)
73 \( 1 + (-0.858 + 0.512i)T^{2} \)
79 \( 1 + (1.37 + 0.446i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.473 - 0.880i)T^{2} \)
89 \( 1 + (-0.900 + 0.433i)T^{2} \)
97 \( 1 + (0.963 - 0.266i)T^{2} \)
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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

−8.662695972274460025674878597199, −7.999128582555856024282930933456, −7.44123753378595212487954331579, −6.60020858110338097281361635571, −5.86059901552661372325245943694, −5.37591025522407005378222345750, −4.51354786981379428198246800406, −3.92336153855779309029060818343, −2.86098932863900642228330175876, −2.13125394339822803214022035733, 1.26654640165442169543037547145, 2.22909351272699460141586647777, 2.97748328274706152541918489670, 3.95668624151254753897285927757, 4.66451311650708862286232612652, 5.29898116845800348651814220408, 5.82579756820351070136885344208, 6.89520017715220884909970236348, 7.69104345038644317302105260943, 8.459477898585582345967093789502

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