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Properties

Label 2-3381-483.206-c0-0-3
Degree $2$
Conductor $3381$
Sign $0.262 - 0.964i$
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  + (0.448 − 0.258i)2-s + (−0.382 − 0.923i)3-s + (−0.366 + 0.633i)4-s + (−0.410 − 0.315i)6-s + 0.896i·8-s + (−0.707 + 0.707i)9-s + (0.725 + 0.0955i)12-s + 0.261i·13-s + (−0.133 − 0.232i)16-s + (−0.133 + 0.5i)18-s + (−0.866 + 0.5i)23-s + (0.828 − 0.343i)24-s + (−0.5 + 0.866i)25-s + (0.0675 + 0.117i)26-s + (0.923 + 0.382i)27-s + ⋯
L(s)  = 1  + (0.448 − 0.258i)2-s + (−0.382 − 0.923i)3-s + (−0.366 + 0.633i)4-s + (−0.410 − 0.315i)6-s + 0.896i·8-s + (−0.707 + 0.707i)9-s + (0.725 + 0.0955i)12-s + 0.261i·13-s + (−0.133 − 0.232i)16-s + (−0.133 + 0.5i)18-s + (−0.866 + 0.5i)23-s + (0.828 − 0.343i)24-s + (−0.5 + 0.866i)25-s + (0.0675 + 0.117i)26-s + (0.923 + 0.382i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.262 - 0.964i$
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: \((2,\ 3381,\ (\ :0),\ 0.262 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7744821151\)
\(L(\frac12)\) \(\approx\) \(0.7744821151\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 \)
23 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 0.261iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + (1.71 + 0.991i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 0.261T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (-1.05 - 0.608i)T + (0.5 + 0.866i)T^{2} \)
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

−8.930655301781600572977999859760, −7.963943605329489864234591740683, −7.53331760290231029449439338094, −6.81719158648081046245960152115, −5.72099099620336251013741435368, −5.32365671806674072399650065264, −4.23676219773865994349077833994, −3.45752828195483571881036992442, −2.46527059376675920595896055669, −1.54203975971757787656929598111, 0.41054962201966368205513010475, 2.10400536019752672941575581778, 3.48207608350948526914421333101, 4.13586186384186685705477321918, 4.80921176961756748154643259372, 5.62971858685855171557178207521, 6.07258006047136713926503416116, 6.87375887995973570904821295401, 7.977944994436604154130324962059, 8.789033522186923402971188880941

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