Properties

Label 2-3381-483.68-c0-0-13
Degree $2$
Conductor $3381$
Sign $-0.835 + 0.549i$
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 + (−0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (1.30 − 0.541i)6-s + (−0.258 − 0.965i)9-s + (−0.991 − 0.130i)12-s − 1.84i·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 + (−1.30 + 2.26i)26-s + (0.923 + 0.382i)27-s + (−0.662 + 0.382i)31-s + (−1.22 + 0.707i)32-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)2-s + (−0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (1.30 − 0.541i)6-s + (−0.258 − 0.965i)9-s + (−0.991 − 0.130i)12-s − 1.84i·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 + (−1.30 + 2.26i)26-s + (0.923 + 0.382i)27-s + (−0.662 + 0.382i)31-s + (−1.22 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.835 + 0.549i$
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: \((2,\ 3381,\ (\ :0),\ -0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2464455454\)
\(L(\frac12)\) \(\approx\) \(0.2464455454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.608 - 0.793i)T \)
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 \( 1 + 1.84iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 2iT - T^{2} \)
73 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
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

−8.606269585929822481257381974774, −8.092779238995882072689545553101, −7.24682534476405637172279798381, −6.15668805560165558483150224699, −5.38412889746415941784334810275, −4.76368769045129678946519317422, −3.42318472562653766491916802414, −2.91307453218250301166296148399, −1.47959158568745335716763277516, −0.26188183844456473860922889908, 1.31857315506665504231014701508, 2.10049388927937453366403394985, 3.65179052307374927676773675046, 4.74918498697115531879625764567, 5.65290351844305126627005469202, 6.54084216721992246152357262898, 6.95747574559628553497284031416, 7.44456808327849142070361756144, 8.417254175395056919415743219143, 8.849184246166155498798356869921

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