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Properties

Label 2-3380-3380.2647-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.999 - 0.0237i$
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  + (0.992 − 0.120i)2-s + (0.970 − 0.239i)4-s + (−0.354 + 0.935i)5-s + (0.935 − 0.354i)8-s + (0.992 − 0.120i)9-s + (−0.239 + 0.970i)10-s + (0.354 − 0.935i)13-s + (0.885 − 0.464i)16-s + (−0.943 − 0.424i)17-s + (0.970 − 0.239i)18-s + (−0.120 + 0.992i)20-s + (−0.748 − 0.663i)25-s + (0.239 − 0.970i)26-s + (1.48 − 0.180i)29-s + (0.822 − 0.568i)32-s + ⋯
L(s)  = 1  + (0.992 − 0.120i)2-s + (0.970 − 0.239i)4-s + (−0.354 + 0.935i)5-s + (0.935 − 0.354i)8-s + (0.992 − 0.120i)9-s + (−0.239 + 0.970i)10-s + (0.354 − 0.935i)13-s + (0.885 − 0.464i)16-s + (−0.943 − 0.424i)17-s + (0.970 − 0.239i)18-s + (−0.120 + 0.992i)20-s + (−0.748 − 0.663i)25-s + (0.239 − 0.970i)26-s + (1.48 − 0.180i)29-s + (0.822 − 0.568i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.999 - 0.0237i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.999 - 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.504514519\)
\(L(\frac12)\) \(\approx\) \(2.504514519\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.992 + 0.120i)T \)
5 \( 1 + (0.354 - 0.935i)T \)
13 \( 1 + (-0.354 + 0.935i)T \)
good3 \( 1 + (-0.992 + 0.120i)T^{2} \)
7 \( 1 + (0.748 - 0.663i)T^{2} \)
11 \( 1 + (0.239 - 0.970i)T^{2} \)
17 \( 1 + (0.943 + 0.424i)T + (0.663 + 0.748i)T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-1.48 + 0.180i)T + (0.970 - 0.239i)T^{2} \)
31 \( 1 + (-0.935 + 0.354i)T^{2} \)
37 \( 1 + (0.527 - 0.764i)T + (-0.354 - 0.935i)T^{2} \)
41 \( 1 + (-0.0950 - 1.57i)T + (-0.992 + 0.120i)T^{2} \)
43 \( 1 + (0.935 + 0.354i)T^{2} \)
47 \( 1 + (-0.885 - 0.464i)T^{2} \)
53 \( 1 + (-0.748 - 0.336i)T + (0.663 + 0.748i)T^{2} \)
59 \( 1 + (0.822 - 0.568i)T^{2} \)
61 \( 1 + (0.470 - 1.24i)T + (-0.748 - 0.663i)T^{2} \)
67 \( 1 + (0.885 + 0.464i)T^{2} \)
71 \( 1 + (0.992 - 0.120i)T^{2} \)
73 \( 1 + (1.98 + 0.241i)T + (0.970 + 0.239i)T^{2} \)
79 \( 1 + (0.885 + 0.464i)T^{2} \)
83 \( 1 + (-0.120 + 0.992i)T^{2} \)
89 \( 1 + (1.35 + 1.35i)T + iT^{2} \)
97 \( 1 + (0.764 + 1.45i)T + (-0.568 + 0.822i)T^{2} \)
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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

−8.641796822413267590046497719164, −7.76193045669979120966606240302, −7.13266648773340477575746683192, −6.50644033179096435544740428154, −5.89310968954700461384947640139, −4.70347097799990374670323312200, −4.25871909865757366937978027041, −3.18532005569032749942865308420, −2.68518870882544833800999668638, −1.37527479584247939126698633201, 1.41381769113400299622869345497, 2.24012109257808561144973031727, 3.66260545398355626466882831138, 4.22901981681451560286444103590, 4.79855593006420545081471582711, 5.60933460141905477515267587481, 6.64374970485458870087381019294, 7.02710726098443897514716459404, 8.001534680043068401024763224427, 8.652473597920161392326566905310

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