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Properties

Label 2-3380-3380.2859-c0-0-1
Degree $2$
Conductor $3380$
Sign $0.817 + 0.575i$
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  + (−0.568 + 0.822i)2-s + (−0.354 − 0.935i)4-s + (−0.970 + 0.239i)5-s + (0.970 + 0.239i)8-s + (−0.568 + 0.822i)9-s + (0.354 − 0.935i)10-s + (−0.970 + 0.239i)13-s + (−0.748 + 0.663i)16-s + (0.447 − 1.81i)17-s + (−0.354 − 0.935i)18-s + (0.568 + 0.822i)20-s + (0.885 − 0.464i)25-s + (0.354 − 0.935i)26-s + (−1.00 + 1.45i)29-s + (−0.120 − 0.992i)32-s + ⋯
L(s)  = 1  + (−0.568 + 0.822i)2-s + (−0.354 − 0.935i)4-s + (−0.970 + 0.239i)5-s + (0.970 + 0.239i)8-s + (−0.568 + 0.822i)9-s + (0.354 − 0.935i)10-s + (−0.970 + 0.239i)13-s + (−0.748 + 0.663i)16-s + (0.447 − 1.81i)17-s + (−0.354 − 0.935i)18-s + (0.568 + 0.822i)20-s + (0.885 − 0.464i)25-s + (0.354 − 0.935i)26-s + (−1.00 + 1.45i)29-s + (−0.120 − 0.992i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4096815029\)
\(L(\frac12)\) \(\approx\) \(0.4096815029\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.568 - 0.822i)T \)
5 \( 1 + (0.970 - 0.239i)T \)
13 \( 1 + (0.970 - 0.239i)T \)
good3 \( 1 + (0.568 - 0.822i)T^{2} \)
7 \( 1 + (-0.885 - 0.464i)T^{2} \)
11 \( 1 + (-0.354 + 0.935i)T^{2} \)
17 \( 1 + (-0.447 + 1.81i)T + (-0.885 - 0.464i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (1.00 - 1.45i)T + (-0.354 - 0.935i)T^{2} \)
31 \( 1 + (-0.970 - 0.239i)T^{2} \)
37 \( 1 + (-0.180 + 1.48i)T + (-0.970 - 0.239i)T^{2} \)
41 \( 1 + (0.764 + 1.45i)T + (-0.568 + 0.822i)T^{2} \)
43 \( 1 + (-0.970 + 0.239i)T^{2} \)
47 \( 1 + (0.748 + 0.663i)T^{2} \)
53 \( 1 + (0.114 - 0.464i)T + (-0.885 - 0.464i)T^{2} \)
59 \( 1 + (0.120 + 0.992i)T^{2} \)
61 \( 1 + (-1.71 + 0.423i)T + (0.885 - 0.464i)T^{2} \)
67 \( 1 + (0.748 + 0.663i)T^{2} \)
71 \( 1 + (0.568 - 0.822i)T^{2} \)
73 \( 1 + (1.13 + 1.64i)T + (-0.354 + 0.935i)T^{2} \)
79 \( 1 + (0.748 + 0.663i)T^{2} \)
83 \( 1 + (-0.568 - 0.822i)T^{2} \)
89 \( 1 + 1.32iT - T^{2} \)
97 \( 1 + (0.180 - 0.159i)T + (0.120 - 0.992i)T^{2} \)
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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

−8.763684153724675445282158161334, −7.67460172703527787630762403505, −7.39528290190556647968869749015, −6.89164778611583546413280954406, −5.56690798291582412873505334923, −5.13863175875343306877702105252, −4.31837352273434179935634970800, −3.12782486534140050175969001746, −2.08717769218330953492603786850, −0.34702761849666964428887127977, 1.06763479834933483538806200641, 2.37346210665640687627198973311, 3.39304838576471690678134981172, 3.91562855213623550339548189105, 4.77838429497766472627225556077, 5.86160877097442309924086205049, 6.84485866456296562604077410095, 7.69402133624803801642512499286, 8.315489997554434550383158798054, 8.690566358010555984006573987760

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