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Properties

Label 2-3311-3311.965-c0-0-0
Degree $2$
Conductor $3311$
Sign $0.868 - 0.495i$
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  + (0.0830 − 1.84i)2-s + (−2.41 − 0.217i)4-s + (−0.978 + 0.207i)7-s + (−0.353 + 2.61i)8-s + (0.599 − 0.800i)9-s + (−0.983 − 0.178i)11-s + (0.303 + 1.82i)14-s + (2.41 + 0.438i)16-s + (−1.42 − 1.17i)18-s + (−0.411 + 1.80i)22-s + (−0.411 − 0.381i)23-s + (−0.998 − 0.0598i)25-s + (2.40 − 0.289i)28-s + (−0.846 + 1.69i)29-s + (0.424 − 1.85i)32-s + ⋯
L(s)  = 1  + (0.0830 − 1.84i)2-s + (−2.41 − 0.217i)4-s + (−0.978 + 0.207i)7-s + (−0.353 + 2.61i)8-s + (0.599 − 0.800i)9-s + (−0.983 − 0.178i)11-s + (0.303 + 1.82i)14-s + (2.41 + 0.438i)16-s + (−1.42 − 1.17i)18-s + (−0.411 + 1.80i)22-s + (−0.411 − 0.381i)23-s + (−0.998 − 0.0598i)25-s + (2.40 − 0.289i)28-s + (−0.846 + 1.69i)29-s + (0.424 − 1.85i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.868 - 0.495i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.868 - 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04933851352\)
\(L(\frac12)\) \(\approx\) \(0.04933851352\)
\(L(1)\) 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 \( 1 + (0.983 + 0.178i)T \)
43 \( 1 + (0.992 - 0.119i)T \)
good2 \( 1 + (-0.0830 + 1.84i)T + (-0.995 - 0.0896i)T^{2} \)
3 \( 1 + (-0.599 + 0.800i)T^{2} \)
5 \( 1 + (0.998 + 0.0598i)T^{2} \)
13 \( 1 + (-0.163 + 0.986i)T^{2} \)
17 \( 1 + (-0.772 + 0.635i)T^{2} \)
19 \( 1 + (0.999 - 0.0299i)T^{2} \)
23 \( 1 + (0.411 + 0.381i)T + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.846 - 1.69i)T + (-0.599 - 0.800i)T^{2} \)
31 \( 1 + (-0.420 - 0.907i)T^{2} \)
37 \( 1 + (1.07 - 0.967i)T + (0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.753 + 0.657i)T^{2} \)
47 \( 1 + (-0.473 + 0.880i)T^{2} \)
53 \( 1 + (-1.03 - 1.67i)T + (-0.447 + 0.894i)T^{2} \)
59 \( 1 + (0.963 - 0.266i)T^{2} \)
61 \( 1 + (-0.420 + 0.907i)T^{2} \)
67 \( 1 + (-0.480 - 0.148i)T + (0.826 + 0.563i)T^{2} \)
71 \( 1 + (0.237 + 1.97i)T + (-0.971 + 0.237i)T^{2} \)
73 \( 1 + (-0.887 - 0.460i)T^{2} \)
79 \( 1 + (-0.0971 + 0.218i)T + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (0.575 + 0.817i)T^{2} \)
89 \( 1 + (0.955 - 0.294i)T^{2} \)
97 \( 1 + (0.691 - 0.722i)T^{2} \)
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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

−9.148010801837019313775979591883, −8.591703694324465401439539729677, −7.52674427174341664898875091189, −6.52839313776122492981574503972, −5.58207829148028446027757570212, −4.78382393249858453561644686316, −3.75751390182728010738820291171, −3.31953877936881287836617174986, −2.44724630526505584863318801270, −1.40856449074170861065675639357, 0.02883430124494093012985665418, 2.21572170058562501875482488518, 3.69923036687526591774746615746, 4.27261960889441630493892697430, 5.39471558191829028787868845926, 5.63408369134444341869462424985, 6.68871997989604861867251155307, 7.19636198628051132267928207872, 7.87431385819991202955895636938, 8.323453519142688609535171197704

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