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Properties

Label 2-3311-3311.1581-c0-0-0
Degree $2$
Conductor $3311$
Sign $-0.995 - 0.0987i$
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  + (1.50 + 1.31i)2-s + (0.400 + 2.95i)4-s + (−0.978 + 0.207i)7-s + (−2.18 + 3.30i)8-s + (0.646 + 0.762i)9-s + (0.963 − 0.266i)11-s + (−1.74 − 0.972i)14-s + (−4.75 + 1.31i)16-s + (−0.0298 + 1.99i)18-s + (1.79 + 0.866i)22-s + (0.270 − 0.184i)23-s + (−0.575 − 0.817i)25-s + (−1.00 − 2.81i)28-s + (−0.441 − 0.953i)29-s + (−5.29 − 2.54i)32-s + ⋯
L(s)  = 1  + (1.50 + 1.31i)2-s + (0.400 + 2.95i)4-s + (−0.978 + 0.207i)7-s + (−2.18 + 3.30i)8-s + (0.646 + 0.762i)9-s + (0.963 − 0.266i)11-s + (−1.74 − 0.972i)14-s + (−4.75 + 1.31i)16-s + (−0.0298 + 1.99i)18-s + (1.79 + 0.866i)22-s + (0.270 − 0.184i)23-s + (−0.575 − 0.817i)25-s + (−1.00 − 2.81i)28-s + (−0.441 − 0.953i)29-s + (−5.29 − 2.54i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.995 - 0.0987i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.995 - 0.0987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.630264410\)
\(L(\frac12)\) \(\approx\) \(2.630264410\)
\(L(1)\) 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 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (-0.337 - 0.941i)T \)
good2 \( 1 + (-1.50 - 1.31i)T + (0.134 + 0.990i)T^{2} \)
3 \( 1 + (-0.646 - 0.762i)T^{2} \)
5 \( 1 + (0.575 + 0.817i)T^{2} \)
13 \( 1 + (-0.873 + 0.486i)T^{2} \)
17 \( 1 + (0.0149 + 0.999i)T^{2} \)
19 \( 1 + (-0.887 + 0.460i)T^{2} \)
23 \( 1 + (-0.270 + 0.184i)T + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (0.441 + 0.953i)T + (-0.646 + 0.762i)T^{2} \)
31 \( 1 + (-0.791 + 0.611i)T^{2} \)
37 \( 1 + (-0.522 + 0.470i)T + (0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.473 - 0.880i)T^{2} \)
47 \( 1 + (0.0448 - 0.998i)T^{2} \)
53 \( 1 + (-1.62 - 1.03i)T + (0.420 + 0.907i)T^{2} \)
59 \( 1 + (0.393 - 0.919i)T^{2} \)
61 \( 1 + (-0.791 - 0.611i)T^{2} \)
67 \( 1 + (0.0896 - 1.19i)T + (-0.988 - 0.149i)T^{2} \)
71 \( 1 + (1.87 - 0.670i)T + (0.772 - 0.635i)T^{2} \)
73 \( 1 + (-0.193 - 0.981i)T^{2} \)
79 \( 1 + (-0.765 + 1.71i)T + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (-0.925 + 0.379i)T^{2} \)
89 \( 1 + (0.0747 + 0.997i)T^{2} \)
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

−8.877128131888350855054479741414, −8.119800325142921782793453418835, −7.34445824018706338634875219525, −6.83516280710779534914743645268, −5.99127202980202415521202782060, −5.68415695198261551259443047946, −4.38670860431283144128754218207, −4.16781408344134021381310541214, −3.12974648044954499446338395765, −2.28391234991680590024076203834, 0.992601431247783988039548838842, 1.93996169871091768407205708867, 3.12935373942659691244380454504, 3.71964929940414548761193529817, 4.21043429416446930580127763693, 5.23516237022195370190288426920, 6.00471702895666708879481268656, 6.72457646353712240482793677779, 7.15760740669426033925768273071, 9.103982569215375408566695676837

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