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Properties

Label 2-3381-483.482-c0-0-15
Degree $2$
Conductor $3381$
Sign $0.878 - 0.477i$
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  + 0.517i·2-s + (0.793 − 0.608i)3-s + 0.732·4-s + (0.315 + 0.410i)6-s + 0.896i·8-s + (0.258 − 0.965i)9-s + (0.580 − 0.445i)12-s + 1.98i·13-s + 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (0.545 + 0.711i)24-s + 25-s − 1.02·26-s + (−0.382 − 0.923i)27-s + ⋯
L(s)  = 1  + 0.517i·2-s + (0.793 − 0.608i)3-s + 0.732·4-s + (0.315 + 0.410i)6-s + 0.896i·8-s + (0.258 − 0.965i)9-s + (0.580 − 0.445i)12-s + 1.98i·13-s + 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (0.545 + 0.711i)24-s + 25-s − 1.02·26-s + (−0.382 − 0.923i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.878 - 0.477i$
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: \((2,\ 3381,\ (\ :0),\ 0.878 - 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.081276209\)
\(L(\frac12)\) \(\approx\) \(2.081276209\)
\(L(1)\) 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 \( 1 - iT \)
good2 \( 1 - 0.517iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.98iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + 0.261iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.98T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.21T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + 1.58iT - T^{2} \)
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

−8.660413675524368027726174798802, −8.015594943307413968039170222926, −7.29385457285408257885536055834, −6.66063815544389229312112801847, −6.29515821300655867920536812645, −5.15600777309541676855446058844, −4.12620185199638378041136430135, −3.20835493851229565475700409045, −2.18266633175742024459786664009, −1.59361213406305004658345028926, 1.26015262008443576264798574961, 2.48491241176584147611796606067, 3.11446806473324696871448850403, 3.66298114999329979301462432538, 4.91236071156606646677761978709, 5.48660662818807675474386248279, 6.70847928307634857003730466827, 7.24373854702043664558780605002, 8.339459373542490014750003375304, 8.490991102833898630069017997414

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