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Properties

Label 2-3381-483.206-c0-0-21
Degree $2$
Conductor $3381$
Sign $-0.0587 + 0.998i$
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  + (1.67 − 0.965i)2-s + (−0.382 + 0.923i)3-s + (1.36 − 2.36i)4-s + (0.252 + 1.91i)6-s − 3.34i·8-s + (−0.707 − 0.707i)9-s + (1.66 + 2.16i)12-s − 1.58i·13-s + (−1.86 − 3.23i)16-s + (−1.86 − 0.500i)18-s + (0.866 − 0.5i)23-s + (3.09 + 1.28i)24-s + (−0.5 + 0.866i)25-s + (−1.53 − 2.65i)26-s + (0.923 − 0.382i)27-s + ⋯
L(s)  = 1  + (1.67 − 0.965i)2-s + (−0.382 + 0.923i)3-s + (1.36 − 2.36i)4-s + (0.252 + 1.91i)6-s − 3.34i·8-s + (−0.707 − 0.707i)9-s + (1.66 + 2.16i)12-s − 1.58i·13-s + (−1.86 − 3.23i)16-s + (−1.86 − 0.500i)18-s + (0.866 − 0.5i)23-s + (3.09 + 1.28i)24-s + (−0.5 + 0.866i)25-s + (−1.53 − 2.65i)26-s + (0.923 − 0.382i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0587 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0587 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.0587 + 0.998i$
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: \((2,\ 3381,\ (\ :0),\ -0.0587 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.835086102\)
\(L(\frac12)\) \(\approx\) \(2.835086102\)
\(L(1)\) 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 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.58iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + (1.05 + 0.608i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.58T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.71 - 0.991i)T + (0.5 + 0.866i)T^{2} \)
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

−9.010752817809943946381552657947, −7.65648311047974192840645976250, −6.69142325697974941218967851016, −5.78104402452264712734769436242, −5.37760816700091997905476161712, −4.76803506109371304917208329962, −3.82117798306596009921147061540, −3.27870059317600354656670014195, −2.51954407514728730137407959642, −1.04925456125721475864993293342, 1.89935509486165289581495182536, 2.66412666504535423503178296992, 3.83042638232095530241452990838, 4.51183829712848004949478376167, 5.32506184013580256135936073968, 6.06329441387011622581087280028, 6.57294928394176090789538361995, 7.22111264710297001960530284388, 7.78277346626915872268946617843, 8.552852995427637829641283170459

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