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Properties

Label 2-3311-3311.349-c0-0-0
Degree $2$
Conductor $3311$
Sign $0.909 - 0.415i$
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  + (0.128 + 0.301i)2-s + (0.616 − 0.645i)4-s + (−0.978 + 0.207i)7-s + (0.580 + 0.217i)8-s + (0.999 − 0.0299i)9-s + (0.0448 + 0.998i)11-s + (−0.188 − 0.267i)14-s + (−0.0309 − 0.688i)16-s + (0.137 + 0.297i)18-s + (−0.295 + 0.142i)22-s + (−0.149 + 1.99i)23-s + (0.873 + 0.486i)25-s + (−0.469 + 0.759i)28-s + (0.0100 − 0.674i)29-s + (0.762 − 0.367i)32-s + ⋯
L(s)  = 1  + (0.128 + 0.301i)2-s + (0.616 − 0.645i)4-s + (−0.978 + 0.207i)7-s + (0.580 + 0.217i)8-s + (0.999 − 0.0299i)9-s + (0.0448 + 0.998i)11-s + (−0.188 − 0.267i)14-s + (−0.0309 − 0.688i)16-s + (0.137 + 0.297i)18-s + (−0.295 + 0.142i)22-s + (−0.149 + 1.99i)23-s + (0.873 + 0.486i)25-s + (−0.469 + 0.759i)28-s + (0.0100 − 0.674i)29-s + (0.762 − 0.367i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.909 - 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.603294240\)
\(L(\frac12)\) \(\approx\) \(1.603294240\)
\(L(1)\) 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 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (0.525 - 0.850i)T \)
good2 \( 1 + (-0.128 - 0.301i)T + (-0.691 + 0.722i)T^{2} \)
3 \( 1 + (-0.999 + 0.0299i)T^{2} \)
5 \( 1 + (-0.873 - 0.486i)T^{2} \)
13 \( 1 + (0.575 - 0.817i)T^{2} \)
17 \( 1 + (0.420 - 0.907i)T^{2} \)
19 \( 1 + (-0.251 - 0.967i)T^{2} \)
23 \( 1 + (0.149 - 1.99i)T + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.0100 + 0.674i)T + (-0.999 - 0.0299i)T^{2} \)
31 \( 1 + (-0.971 - 0.237i)T^{2} \)
37 \( 1 + (-1.24 + 1.11i)T + (0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.983 - 0.178i)T^{2} \)
47 \( 1 + (0.963 + 0.266i)T^{2} \)
53 \( 1 + (0.0639 - 0.0629i)T + (0.0149 - 0.999i)T^{2} \)
59 \( 1 + (-0.753 + 0.657i)T^{2} \)
61 \( 1 + (-0.971 + 0.237i)T^{2} \)
67 \( 1 + (-0.319 + 0.217i)T + (0.365 - 0.930i)T^{2} \)
71 \( 1 + (1.69 + 1.04i)T + (0.447 + 0.894i)T^{2} \)
73 \( 1 + (0.599 + 0.800i)T^{2} \)
79 \( 1 + (-0.692 + 1.55i)T + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (-0.280 + 0.959i)T^{2} \)
89 \( 1 + (0.826 + 0.563i)T^{2} \)
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

−9.190982680489999979507365927823, −7.64319428426862523010721790823, −7.40787297243926116136247495516, −6.59293001296447464867776261470, −5.99558361760469667172367696901, −5.14457852570503368003979840702, −4.33019156730972111324291009504, −3.33348535885170366528542305483, −2.23281871210371131795094770233, −1.32787642854872626263131824116, 1.06399193300827268716943363617, 2.47175330295112495555599734065, 3.12141927090147962482453095307, 3.97981054288558812182597282157, 4.66009145296727427091604338802, 5.97192096398680694919380703320, 6.74009543346644048188238157196, 6.99130601840978290085771695863, 8.122759596704044456047569708275, 8.603107375087312851855694243769

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