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Properties

Label 2-3380-20.19-c0-0-0
Degree $2$
Conductor $3380$
Sign $-i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
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  − i·2-s − 4-s + i·5-s + i·8-s − 9-s + 10-s + 16-s + i·18-s − i·20-s − 25-s − 2·29-s − i·32-s + 36-s + 2i·37-s − 40-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·5-s + i·8-s − 9-s + 10-s + 16-s + i·18-s − i·20-s − 25-s − 2·29-s − i·32-s + 36-s + 2i·37-s − 40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4318924400\)
\(L(\frac12)\) \(\approx\) \(0.4318924400\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 - iT \)
13 \( 1 \)
good3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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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

−9.201930105867218991140456352657, −8.276521512889493294865449810392, −7.72591198093101937378975496987, −6.67786668396385405165433782549, −5.85645953060586057364875749718, −5.14774543951793505635702552605, −4.06422033194863464548550014437, −3.25041925647795689365764442802, −2.67270663613797552839996682190, −1.64682921709922356036289994969, 0.24837756185276232671804530804, 1.80775891378819593112097772001, 3.30477937222963141198070206861, 4.15348870231783225099970153762, 4.99603942625448563993864991974, 5.69014398837485240440595787605, 6.12086284788632506879586593185, 7.30888891040593399491054240059, 7.82697526295832447897729797978, 8.580781611809621146621467524484

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