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Properties

Label 2-3920-980.319-c0-0-1
Degree $2$
Conductor $3920$
Sign $0.981 + 0.191i$
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  + (−0.108 − 0.277i)3-s + (0.988 + 0.149i)5-s + (0.563 + 0.826i)7-s + (0.667 − 0.619i)9-s + (−0.0663 − 0.290i)15-s + (0.167 − 0.246i)21-s + (−0.139 − 1.85i)23-s + (0.955 + 0.294i)25-s + (−0.513 − 0.247i)27-s + (−0.134 + 0.0648i)29-s + (0.433 + 0.900i)35-s + (0.914 − 1.14i)41-s + (0.848 + 1.06i)43-s + (0.752 − 0.513i)45-s + (−1.49 + 0.460i)47-s + ⋯
L(s)  = 1  + (−0.108 − 0.277i)3-s + (0.988 + 0.149i)5-s + (0.563 + 0.826i)7-s + (0.667 − 0.619i)9-s + (−0.0663 − 0.290i)15-s + (0.167 − 0.246i)21-s + (−0.139 − 1.85i)23-s + (0.955 + 0.294i)25-s + (−0.513 − 0.247i)27-s + (−0.134 + 0.0648i)29-s + (0.433 + 0.900i)35-s + (0.914 − 1.14i)41-s + (0.848 + 1.06i)43-s + (0.752 − 0.513i)45-s + (−1.49 + 0.460i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.981 + 0.191i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.981 + 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.717041468\)
\(L(\frac12)\) \(\approx\) \(1.717041468\)
\(L(1)\) 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 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (-0.563 - 0.826i)T \)
good3 \( 1 + (0.108 + 0.277i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.139 + 1.85i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (-0.914 + 1.14i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.848 - 1.06i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (1.49 - 0.460i)T + (0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (0.603 + 0.411i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.250 - 1.09i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
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

−8.715829316729858205326497393049, −7.927906120687879567074959418443, −7.01763744115686333355109052601, −6.32152695025625090240819699457, −5.83116588813177110093688002976, −4.92011533135067796928436461330, −4.18387520035653018168326722434, −2.88086226989163885718925986382, −2.16166504739016561145863442079, −1.20750699315360442575544181836, 1.33713506877218378811132725586, 1.98913929221880840692042844519, 3.28083038064111247766374586390, 4.24763657828207484479973485519, 4.92537263947991714812230755315, 5.57710297459758353705137865101, 6.41490330838188327378979752487, 7.38012019965487071634071233489, 7.73890371590323992566014451687, 8.739769207238653677844860204573

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