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Properties

Label 2-3381-483.482-c0-0-22
Degree $2$
Conductor $3381$
Sign $-0.972 + 0.233i$
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  − 1.41i·2-s + (0.923 − 0.382i)3-s − 1.00·4-s + (−0.541 − 1.30i)6-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)12-s − 0.765i·13-s − 0.999·16-s + (−1.00 − i)18-s − i·23-s + 25-s − 1.08·26-s + (0.382 − 0.923i)27-s + 1.84i·31-s + 1.41i·32-s + ⋯
L(s)  = 1  − 1.41i·2-s + (0.923 − 0.382i)3-s − 1.00·4-s + (−0.541 − 1.30i)6-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)12-s − 0.765i·13-s − 0.999·16-s + (−1.00 − i)18-s − i·23-s + 25-s − 1.08·26-s + (0.382 − 0.923i)27-s + 1.84i·31-s + 1.41i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.972 + 0.233i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (3380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ -0.972 + 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.734601990\)
\(L(\frac12)\) \(\approx\) \(1.734601990\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 \)
23 \( 1 + iT \)
good2 \( 1 + 1.41iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 0.765iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.84iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.765T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 0.765T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 - 1.84iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
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

−8.753228811183593212910275181989, −7.921061635016528623060460266361, −7.01975949435002868172316675109, −6.40246504438196727514117995697, −5.05934392444512838441553851025, −4.26160364775161721727916737979, −3.25213482432495438759603118624, −2.89610536498572168842851833290, −1.90244534019437598780644814806, −0.960705407554033885350667234903, 1.80508170378771722012171428637, 2.83301132027076209522150946685, 3.94458697219148524002949471647, 4.63006185745894415018139296042, 5.42498502380004612584396979831, 6.28240433635837408768435482282, 7.05963629361279602150327858637, 7.64526836111213968397864204174, 8.216882095027301463680511534515, 9.086701100533405658269068786721

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