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Properties

Label 2-3381-483.206-c0-0-4
Degree $2$
Conductor $3381$
Sign $0.530 - 0.847i$
Analytic cond. $1.68733$
Root an. cond. $1.29897$
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  + (−1.22 + 0.707i)2-s + (0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (0.541 + 1.30i)6-s + (−0.965 − 0.258i)9-s + (−0.793 − 0.608i)12-s + 0.765i·13-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)18-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)25-s + (−0.541 − 0.937i)26-s + (−0.382 + 0.923i)27-s + (1.60 + 0.923i)31-s + (−1.22 − 0.707i)32-s + ⋯
L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (0.541 + 1.30i)6-s + (−0.965 − 0.258i)9-s + (−0.793 − 0.608i)12-s + 0.765i·13-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)18-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)25-s + (−0.541 − 0.937i)26-s + (−0.382 + 0.923i)27-s + (1.60 + 0.923i)31-s + (−1.22 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (2138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ 0.530 - 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5498716335\)
\(L(\frac12)\) \(\approx\) \(0.5498716335\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.130 + 0.991i)T \)
7 \( 1 \)
23 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 0.765iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.868574192426608887073696670289, −7.959894114142122034986152441198, −7.65597193537825048266188680248, −6.83048851363085511957821532923, −6.30754053257238025218593400708, −5.58205550215009410070348134395, −4.29465047594632348171763424856, −3.19624699016846838537836044113, −1.98477377586055022306373421569, −1.08455722672001025881564583674, 0.56862006233549971796611074455, 2.17080423212485545605995002342, 2.82569968937345352262689831457, 3.84582262370025779165608701189, 4.70995059557042497305623134067, 5.61673915422475700488257398468, 6.39612372337482093408108225703, 7.77638432595850907943343445078, 8.159166251676356188435248357513, 8.757842860913968817086092417339

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