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Properties

Label 2-3381-483.68-c0-0-12
Degree $2$
Conductor $3381$
Sign $-0.0375 - 0.999i$
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 + (0.793 + 0.608i)3-s + (0.499 + 0.866i)4-s + (0.541 + 1.30i)6-s + (0.258 + 0.965i)9-s + (−0.130 + 0.991i)12-s + 0.765i·13-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)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 + (0.793 + 0.608i)3-s + (0.499 + 0.866i)4-s + (0.541 + 1.30i)6-s + (0.258 + 0.965i)9-s + (−0.130 + 0.991i)12-s + 0.765i·13-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0375 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0375 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.0375 - 0.999i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ -0.0375 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.146685684\)
\(L(\frac12)\) \(\approx\) \(3.146685684\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.793 - 0.608i)T \)
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

−9.048602213439632813060248012405, −8.067802104235017007568078319651, −7.34202716615135478495410070192, −6.73688891599661438110158272263, −5.78892262084931309303521049873, −5.06258409668236861991689294713, −4.40149670184181195387772788262, −3.70187300200034831129824439677, −2.99651174891056084390301288064, −1.82888092084005522981001804448, 1.31815751956786027218368022342, 2.34966855760622676340947453383, 3.02611717910714033541637907813, 3.75629369924853365167382514881, 4.51141882645759724213493172838, 5.56611817034502003501629617356, 6.06003885943287707553534604546, 7.23599000493361023243461191500, 7.70024907375180303754689721636, 8.653712172595084076407282818988

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