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Properties

Label 2-3311-3311.503-c0-0-0
Degree $2$
Conductor $3311$
Sign $-0.924 - 0.380i$
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  + (0.607 + 1.42i)2-s + (−0.958 + 1.00i)4-s + (0.669 + 0.743i)7-s + (−0.559 − 0.209i)8-s + (−0.525 − 0.850i)9-s + (0.0448 + 0.998i)11-s + (−0.649 + 1.40i)14-s + (0.0208 + 0.463i)16-s + (0.889 − 1.26i)18-s + (−1.39 + 0.670i)22-s + (0.739 + 0.503i)23-s + (−0.0149 − 0.999i)25-s + (−1.38 − 0.0414i)28-s + (−1.12 + 0.629i)29-s + (−1.18 + 0.570i)32-s + ⋯
L(s)  = 1  + (0.607 + 1.42i)2-s + (−0.958 + 1.00i)4-s + (0.669 + 0.743i)7-s + (−0.559 − 0.209i)8-s + (−0.525 − 0.850i)9-s + (0.0448 + 0.998i)11-s + (−0.649 + 1.40i)14-s + (0.0208 + 0.463i)16-s + (0.889 − 1.26i)18-s + (−1.39 + 0.670i)22-s + (0.739 + 0.503i)23-s + (−0.0149 − 0.999i)25-s + (−1.38 − 0.0414i)28-s + (−1.12 + 0.629i)29-s + (−1.18 + 0.570i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.924 - 0.380i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.924 - 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.737451449\)
\(L(\frac12)\) \(\approx\) \(1.737451449\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (-0.999 - 0.0299i)T \)
good2 \( 1 + (-0.607 - 1.42i)T + (-0.691 + 0.722i)T^{2} \)
3 \( 1 + (0.525 + 0.850i)T^{2} \)
5 \( 1 + (0.0149 + 0.999i)T^{2} \)
13 \( 1 + (0.420 + 0.907i)T^{2} \)
17 \( 1 + (0.575 + 0.817i)T^{2} \)
19 \( 1 + (-0.712 + 0.701i)T^{2} \)
23 \( 1 + (-0.739 - 0.503i)T + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (1.12 - 0.629i)T + (0.525 - 0.850i)T^{2} \)
31 \( 1 + (0.280 + 0.959i)T^{2} \)
37 \( 1 + (-0.347 - 1.63i)T + (-0.913 + 0.406i)T^{2} \)
41 \( 1 + (0.983 - 0.178i)T^{2} \)
47 \( 1 + (0.963 + 0.266i)T^{2} \)
53 \( 1 + (0.0225 + 0.0868i)T + (-0.873 + 0.486i)T^{2} \)
59 \( 1 + (-0.753 + 0.657i)T^{2} \)
61 \( 1 + (0.280 - 0.959i)T^{2} \)
67 \( 1 + (0.141 + 1.88i)T + (-0.988 + 0.149i)T^{2} \)
71 \( 1 + (-0.0243 + 0.813i)T + (-0.998 - 0.0598i)T^{2} \)
73 \( 1 + (-0.992 + 0.119i)T^{2} \)
79 \( 1 + (0.0595 - 0.00625i)T + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.971 - 0.237i)T^{2} \)
89 \( 1 + (0.0747 - 0.997i)T^{2} \)
97 \( 1 + (0.550 + 0.834i)T^{2} \)
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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

−8.989773793123943874861981487907, −8.127757302098752132544201726889, −7.59967365169098097720972411886, −6.73400744395668000432990039299, −6.19582376428329738354388171564, −5.38889923498168906832456033748, −4.83067954980909315317424335698, −4.02956861310533174853717183324, −2.94262748519359218799763321772, −1.69560198356264691532717857287, 0.893707151201485122832732877616, 1.99408189112341037388706083233, 2.81202753626306935462736901340, 3.73366805735200317907626583857, 4.35394967946216851473402891058, 5.28506164366032476029210226766, 5.76898019577487006832298552408, 7.18532318500266118112930238389, 7.73882838640639675266688291884, 8.660199929723498396468596024030

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