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Properties

Label 2-3920-980.219-c0-0-1
Degree $2$
Conductor $3920$
Sign $0.860 - 0.509i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (1.29 − 0.400i)3-s + (0.733 + 0.680i)5-s + (−0.149 + 0.988i)7-s + (0.702 − 0.479i)9-s + (1.22 + 0.590i)15-s + (0.202 + 1.34i)21-s + (−0.215 − 0.548i)23-s + (0.0747 + 0.997i)25-s + (−0.126 + 0.158i)27-s + (0.455 + 0.571i)29-s + (−0.781 + 0.623i)35-s + (0.367 − 1.61i)41-s + (−0.250 − 1.09i)43-s + (0.841 + 0.126i)45-s + (−0.145 + 1.94i)47-s + ⋯
L(s)  = 1  + (1.29 − 0.400i)3-s + (0.733 + 0.680i)5-s + (−0.149 + 0.988i)7-s + (0.702 − 0.479i)9-s + (1.22 + 0.590i)15-s + (0.202 + 1.34i)21-s + (−0.215 − 0.548i)23-s + (0.0747 + 0.997i)25-s + (−0.126 + 0.158i)27-s + (0.455 + 0.571i)29-s + (−0.781 + 0.623i)35-s + (0.367 − 1.61i)41-s + (−0.250 − 1.09i)43-s + (0.841 + 0.126i)45-s + (−0.145 + 1.94i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.860 - 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.246510768\)
\(L(\frac12)\) \(\approx\) \(2.246510768\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.733 - 0.680i)T \)
7 \( 1 + (0.149 - 0.988i)T \)
good3 \( 1 + (-1.29 + 0.400i)T + (0.826 - 0.563i)T^{2} \)
11 \( 1 + (-0.365 - 0.930i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.215 + 0.548i)T + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (-0.455 - 0.571i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.955 + 0.294i)T^{2} \)
41 \( 1 + (-0.367 + 1.61i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.250 + 1.09i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.145 - 1.94i)T + (-0.988 - 0.149i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (-1.88 + 0.284i)T + (0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.268 + 0.129i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.603 + 0.411i)T + (0.365 - 0.930i)T^{2} \)
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

−8.799810234747062402397056683998, −8.071797691968484725155783688687, −7.29367736105714059720183632247, −6.59894611240001555464417184503, −5.85372519849964863455384854732, −5.07781329887180632046051714613, −3.79672197863260336875754573998, −2.96154874219959541420015450307, −2.41618625592160156889095532806, −1.69428195520167561546288280164, 1.16812938641592592971191932750, 2.23622265487504911933505540274, 3.11361452976216379169601547680, 3.98493336440861036757411137447, 4.57044867447830781386574728309, 5.53271210750142227526792432247, 6.44216741780414200116510017777, 7.25767920464669032552574731110, 8.190762376626930590543905108046, 8.437725251026396138784899809749

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