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Properties

Label 2-3311-3311.524-c0-0-0
Degree $2$
Conductor $3311$
Sign $-0.990 + 0.136i$
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  + (−1.41 + 1.23i)2-s + (0.336 − 2.48i)4-s + (−0.309 − 0.951i)7-s + (1.55 + 2.35i)8-s + (−0.983 − 0.178i)9-s + (−0.753 − 0.657i)11-s + (1.60 + 0.960i)14-s + (−2.67 − 0.738i)16-s + (1.60 − 0.960i)18-s + 1.87·22-s + (−0.992 + 0.478i)23-s + (0.995 + 0.0896i)25-s + (−2.46 + 0.447i)28-s + (−1.95 + 0.176i)29-s + (2.13 − 1.03i)32-s + ⋯
L(s)  = 1  + (−1.41 + 1.23i)2-s + (0.336 − 2.48i)4-s + (−0.309 − 0.951i)7-s + (1.55 + 2.35i)8-s + (−0.983 − 0.178i)9-s + (−0.753 − 0.657i)11-s + (1.60 + 0.960i)14-s + (−2.67 − 0.738i)16-s + (1.60 − 0.960i)18-s + 1.87·22-s + (−0.992 + 0.478i)23-s + (0.995 + 0.0896i)25-s + (−2.46 + 0.447i)28-s + (−1.95 + 0.176i)29-s + (2.13 − 1.03i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.990 + 0.136i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.990 + 0.136i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08313766541\)
\(L(\frac12)\) \(\approx\) \(0.08313766541\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (-0.473 - 0.880i)T \)
good2 \( 1 + (1.41 - 1.23i)T + (0.134 - 0.990i)T^{2} \)
3 \( 1 + (0.983 + 0.178i)T^{2} \)
5 \( 1 + (-0.995 - 0.0896i)T^{2} \)
13 \( 1 + (0.858 - 0.512i)T^{2} \)
17 \( 1 + (0.858 + 0.512i)T^{2} \)
19 \( 1 + (0.0448 + 0.998i)T^{2} \)
23 \( 1 + (0.992 - 0.478i)T + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (1.95 - 0.176i)T + (0.983 - 0.178i)T^{2} \)
31 \( 1 + (-0.134 + 0.990i)T^{2} \)
37 \( 1 + (-1.58 + 0.515i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.473 + 0.880i)T^{2} \)
47 \( 1 + (0.0448 + 0.998i)T^{2} \)
53 \( 1 + (-0.00402 - 0.0896i)T + (-0.995 + 0.0896i)T^{2} \)
59 \( 1 + (0.393 + 0.919i)T^{2} \)
61 \( 1 + (-0.134 - 0.990i)T^{2} \)
67 \( 1 + (1.35 + 0.653i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.209 - 1.15i)T + (-0.936 + 0.351i)T^{2} \)
73 \( 1 + (-0.753 - 0.657i)T^{2} \)
79 \( 1 + (1.03 - 1.42i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.134 + 0.990i)T^{2} \)
89 \( 1 + (-0.900 + 0.433i)T^{2} \)
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

−9.110880212742248514133535455861, −8.328936937722882297169920426692, −7.70950314330416777626142679821, −7.27513490999847031041732583918, −6.23587111813665254915197907997, −5.86124658883232456229965078281, −5.04445158955425255535390048889, −3.77877462696650305403855402836, −2.56581231443729353405143039893, −1.11955467859726860729659862579, 0.086712466742934185521745795123, 1.84249413947055789618083151216, 2.53724634783783358422065190216, 3.12069954224527655711264399836, 4.25155141779526164690089479773, 5.40755721732857875124879604749, 6.26656950198448067391197790075, 7.43340909517924958009324408584, 7.934705660352943030993374945872, 8.721778003620339463006791625742

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