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Properties

Label 2-3311-3311.321-c0-0-0
Degree $2$
Conductor $3311$
Sign $-0.991 - 0.129i$
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  + (−1.87 + 0.702i)2-s + (2.25 − 1.97i)4-s + (−0.104 + 0.994i)7-s + (−1.89 + 3.51i)8-s + (−0.420 − 0.907i)9-s + (−0.134 + 0.990i)11-s + (−0.502 − 1.93i)14-s + (0.668 − 4.93i)16-s + (1.42 + 1.40i)18-s + (−0.444 − 1.94i)22-s + (1.23 + 0.381i)23-s + (−0.887 + 0.460i)25-s + (1.72 + 2.45i)28-s + (−1.47 + 0.940i)29-s + (1.32 + 5.81i)32-s + ⋯
L(s)  = 1  + (−1.87 + 0.702i)2-s + (2.25 − 1.97i)4-s + (−0.104 + 0.994i)7-s + (−1.89 + 3.51i)8-s + (−0.420 − 0.907i)9-s + (−0.134 + 0.990i)11-s + (−0.502 − 1.93i)14-s + (0.668 − 4.93i)16-s + (1.42 + 1.40i)18-s + (−0.444 − 1.94i)22-s + (1.23 + 0.381i)23-s + (−0.887 + 0.460i)25-s + (1.72 + 2.45i)28-s + (−1.47 + 0.940i)29-s + (1.32 + 5.81i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.991 - 0.129i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.991 - 0.129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2553142136\)
\(L(\frac12)\) \(\approx\) \(0.2553142136\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (0.575 + 0.817i)T \)
good2 \( 1 + (1.87 - 0.702i)T + (0.753 - 0.657i)T^{2} \)
3 \( 1 + (0.420 + 0.907i)T^{2} \)
5 \( 1 + (0.887 - 0.460i)T^{2} \)
13 \( 1 + (0.251 - 0.967i)T^{2} \)
17 \( 1 + (0.712 - 0.701i)T^{2} \)
19 \( 1 + (-0.971 - 0.237i)T^{2} \)
23 \( 1 + (-1.23 - 0.381i)T + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (1.47 - 0.940i)T + (0.420 - 0.907i)T^{2} \)
31 \( 1 + (0.946 + 0.323i)T^{2} \)
37 \( 1 + (-0.145 - 0.326i)T + (-0.669 + 0.743i)T^{2} \)
41 \( 1 + (0.858 + 0.512i)T^{2} \)
47 \( 1 + (0.691 - 0.722i)T^{2} \)
53 \( 1 + (0.0752 + 0.257i)T + (-0.842 + 0.538i)T^{2} \)
59 \( 1 + (0.550 - 0.834i)T^{2} \)
61 \( 1 + (0.946 - 0.323i)T^{2} \)
67 \( 1 + (-0.655 - 0.608i)T + (0.0747 + 0.997i)T^{2} \)
71 \( 1 + (1.21 - 0.855i)T + (0.337 - 0.941i)T^{2} \)
73 \( 1 + (0.772 - 0.635i)T^{2} \)
79 \( 1 + (0.340 + 1.59i)T + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.193 - 0.981i)T^{2} \)
89 \( 1 + (-0.733 + 0.680i)T^{2} \)
97 \( 1 + (-0.983 - 0.178i)T^{2} \)
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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

−9.016565406508622678758384323089, −8.707786123359243274093243202156, −7.69312992189604025444611002932, −7.13805026823871506376532189732, −6.45984200530630478472381545935, −5.65799003528022989932275547248, −5.14464036053420083012410360135, −3.30275806114107246798619131578, −2.30304576207232688005572080017, −1.42677620646055746490269254798, 0.28437177316875692390713447733, 1.50012214421097324192668885555, 2.54259516938710266343847301804, 3.31128071833829072875538552898, 4.21166466659101302820305403387, 5.71856599046118221839987965301, 6.58955104754959768522327948121, 7.38030005889715979621602594669, 7.955477301196338627164233669454, 8.413274091081650541425937173290

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