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Properties

Label 2-3311-3311.1273-c0-0-0
Degree $2$
Conductor $3311$
Sign $0.607 + 0.794i$
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  + (−0.673 − 0.588i)2-s + (−0.0267 − 0.197i)4-s + (0.669 + 0.743i)7-s + (−0.591 + 0.895i)8-s + (0.337 − 0.941i)9-s + (0.963 − 0.266i)11-s + (−0.0133 − 0.894i)14-s + (0.733 − 0.202i)16-s + (−0.781 + 0.435i)18-s + (−0.806 − 0.388i)22-s + (0.115 + 1.54i)23-s + (−0.420 + 0.907i)25-s + (0.129 − 0.152i)28-s + (1.15 − 1.63i)29-s + (0.353 + 0.170i)32-s + ⋯
L(s)  = 1  + (−0.673 − 0.588i)2-s + (−0.0267 − 0.197i)4-s + (0.669 + 0.743i)7-s + (−0.591 + 0.895i)8-s + (0.337 − 0.941i)9-s + (0.963 − 0.266i)11-s + (−0.0133 − 0.894i)14-s + (0.733 − 0.202i)16-s + (−0.781 + 0.435i)18-s + (−0.806 − 0.388i)22-s + (0.115 + 1.54i)23-s + (−0.420 + 0.907i)25-s + (0.129 − 0.152i)28-s + (1.15 − 1.63i)29-s + (0.353 + 0.170i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.607 + 0.794i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.607 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004805466\)
\(L(\frac12)\) \(\approx\) \(1.004805466\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (-0.646 + 0.762i)T \)
good2 \( 1 + (0.673 + 0.588i)T + (0.134 + 0.990i)T^{2} \)
3 \( 1 + (-0.337 + 0.941i)T^{2} \)
5 \( 1 + (0.420 - 0.907i)T^{2} \)
13 \( 1 + (0.0149 - 0.999i)T^{2} \)
17 \( 1 + (-0.873 - 0.486i)T^{2} \)
19 \( 1 + (0.842 + 0.538i)T^{2} \)
23 \( 1 + (-0.115 - 1.54i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (-1.15 + 1.63i)T + (-0.337 - 0.941i)T^{2} \)
31 \( 1 + (0.925 + 0.379i)T^{2} \)
37 \( 1 + (-0.146 - 0.687i)T + (-0.913 + 0.406i)T^{2} \)
41 \( 1 + (0.473 - 0.880i)T^{2} \)
47 \( 1 + (0.0448 - 0.998i)T^{2} \)
53 \( 1 + (1.71 - 0.888i)T + (0.575 - 0.817i)T^{2} \)
59 \( 1 + (0.393 - 0.919i)T^{2} \)
61 \( 1 + (0.925 - 0.379i)T^{2} \)
67 \( 1 + (-1.64 - 1.11i)T + (0.365 + 0.930i)T^{2} \)
71 \( 1 + (0.620 + 0.525i)T + (0.163 + 0.986i)T^{2} \)
73 \( 1 + (0.946 + 0.323i)T^{2} \)
79 \( 1 + (-1.51 + 0.159i)T + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.791 + 0.611i)T^{2} \)
89 \( 1 + (0.826 - 0.563i)T^{2} \)
97 \( 1 + (-0.936 + 0.351i)T^{2} \)
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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

−8.870505008499644406627487778691, −8.247527308831084545996015506609, −7.37666802260720491864479122112, −6.27194827478082425018072967352, −5.83609878450800509247166246147, −4.89888975876574527448716973210, −3.89528637111581985422925233671, −2.93656867623139871286340143198, −1.82023959778774996262002874448, −1.07206090678834801327697315172, 1.03925081926698751182129038198, 2.26705743405072947964814084850, 3.53123715754520828104543487736, 4.41639746216011068584572594894, 4.93201449916090948554437269792, 6.36477299809241994012865147686, 6.77273227563002114682779490398, 7.57233798769972282594069429372, 8.136061765559266621421786480861, 8.687782795718825388976707097317

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