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Properties

Label 2-3311-3311.1448-c0-0-0
Degree $2$
Conductor $3311$
Sign $0.959 - 0.282i$
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  + (−1.71 − 0.311i)2-s + (1.92 + 0.720i)4-s + (−0.978 − 0.207i)7-s + (−1.57 − 0.941i)8-s + (−0.887 − 0.460i)9-s + (−0.753 − 0.657i)11-s + (1.61 + 0.662i)14-s + (0.871 + 0.761i)16-s + (1.38 + 1.06i)18-s + (1.08 + 1.36i)22-s + (−0.420 + 1.07i)23-s + (0.280 + 0.959i)25-s + (−1.72 − 1.10i)28-s + (−1.38 + 0.337i)29-s + (−0.114 − 0.143i)32-s + ⋯
L(s)  = 1  + (−1.71 − 0.311i)2-s + (1.92 + 0.720i)4-s + (−0.978 − 0.207i)7-s + (−1.57 − 0.941i)8-s + (−0.887 − 0.460i)9-s + (−0.753 − 0.657i)11-s + (1.61 + 0.662i)14-s + (0.871 + 0.761i)16-s + (1.38 + 1.06i)18-s + (1.08 + 1.36i)22-s + (−0.420 + 1.07i)23-s + (0.280 + 0.959i)25-s + (−1.72 − 1.10i)28-s + (−1.38 + 0.337i)29-s + (−0.114 − 0.143i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.959 - 0.282i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1448, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.959 - 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2995729927\)
\(L(\frac12)\) \(\approx\) \(0.2995729927\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.978 + 0.207i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (-0.842 - 0.538i)T \)
good2 \( 1 + (1.71 + 0.311i)T + (0.936 + 0.351i)T^{2} \)
3 \( 1 + (0.887 + 0.460i)T^{2} \)
5 \( 1 + (-0.280 - 0.959i)T^{2} \)
13 \( 1 + (-0.925 + 0.379i)T^{2} \)
17 \( 1 + (0.791 - 0.611i)T^{2} \)
19 \( 1 + (0.599 + 0.800i)T^{2} \)
23 \( 1 + (0.420 - 1.07i)T + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (1.38 - 0.337i)T + (0.887 - 0.460i)T^{2} \)
31 \( 1 + (0.772 - 0.635i)T^{2} \)
37 \( 1 + (-0.133 - 0.119i)T + (0.104 + 0.994i)T^{2} \)
41 \( 1 + (-0.963 + 0.266i)T^{2} \)
47 \( 1 + (0.393 - 0.919i)T^{2} \)
53 \( 1 + (-1.49 + 0.179i)T + (0.971 - 0.237i)T^{2} \)
59 \( 1 + (-0.473 + 0.880i)T^{2} \)
61 \( 1 + (0.772 + 0.635i)T^{2} \)
67 \( 1 + (-1.97 - 0.297i)T + (0.955 + 0.294i)T^{2} \)
71 \( 1 + (-1.07 + 1.67i)T + (-0.420 - 0.907i)T^{2} \)
73 \( 1 + (0.646 + 0.762i)T^{2} \)
79 \( 1 + (0.437 + 0.983i)T + (-0.669 + 0.743i)T^{2} \)
83 \( 1 + (-0.163 + 0.986i)T^{2} \)
89 \( 1 + (-0.988 + 0.149i)T^{2} \)
97 \( 1 + (0.995 - 0.0896i)T^{2} \)
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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

−9.091839664092251729461958750852, −8.228370909426879431139665068819, −7.58632126692947433200755389335, −6.92097935967415213509757981714, −6.03145647680273304822367262126, −5.39057236892250152360186395685, −3.64224304192308659591621289827, −3.09517554204434752637680635955, −2.13594990232543080248002085548, −0.75976940101413194929045880748, 0.45492935297896685515071441205, 2.24540717471681405129490473999, 2.58079747585039832552362692895, 4.01885889739806810003604228857, 5.35604774274111035097234137646, 6.02058744064160265103023389393, 6.82087909204586344671052062939, 7.42649893962241413717632500912, 8.281106993368750097310371642716, 8.626879080787729246588517216792

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