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Properties

Label 2-3920-980.499-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  + (1.84 + 0.277i)3-s + (−0.365 − 0.930i)5-s + (0.997 − 0.0747i)7-s + (2.35 + 0.726i)9-s + (−0.414 − 1.81i)15-s + (1.85 + 0.139i)21-s + (−0.246 + 0.167i)23-s + (−0.733 + 0.680i)25-s + (2.45 + 1.18i)27-s + (−1.48 + 0.716i)29-s + (−0.433 − 0.900i)35-s + (−1.19 + 1.49i)41-s + (−0.367 − 0.460i)43-s + (−0.184 − 2.45i)45-s + (−1.14 − 1.06i)47-s + ⋯
L(s)  = 1  + (1.84 + 0.277i)3-s + (−0.365 − 0.930i)5-s + (0.997 − 0.0747i)7-s + (2.35 + 0.726i)9-s + (−0.414 − 1.81i)15-s + (1.85 + 0.139i)21-s + (−0.246 + 0.167i)23-s + (−0.733 + 0.680i)25-s + (2.45 + 1.18i)27-s + (−1.48 + 0.716i)29-s + (−0.433 − 0.900i)35-s + (−1.19 + 1.49i)41-s + (−0.367 − 0.460i)43-s + (−0.184 − 2.45i)45-s + (−1.14 − 1.06i)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: $\chi_{3920} (3439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.981 + 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.603551923\)
\(L(\frac12)\) \(\approx\) \(2.603551923\)
\(L(1)\) 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 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (-0.997 + 0.0747i)T \)
good3 \( 1 + (-1.84 - 0.277i)T + (0.955 + 0.294i)T^{2} \)
11 \( 1 + (-0.826 + 0.563i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.246 - 0.167i)T + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (1.48 - 0.716i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (1.19 - 1.49i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.367 + 0.460i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (1.14 + 1.06i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.147 + 1.97i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.443 - 1.94i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-1.57 - 0.487i)T + (0.826 + 0.563i)T^{2} \)
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

−8.511649867443095482591020521985, −8.029006411404705555126925863691, −7.63961765500033187997595891714, −6.69877361271441456522810492630, −5.19988926767350554356229827910, −4.78842123748015075168481342938, −3.83918562587801451205741819334, −3.35889365869220800035127539196, −2.05533025984916761279385400205, −1.50894754381764573390088109550, 1.63250811442692923365637826157, 2.30672404267607447348123258823, 3.13534936881634217172229259402, 3.86962375646484779367262247053, 4.54322942855831142027423992639, 5.80549264801520070020772142617, 6.84459582475502283175840771983, 7.49795117210360886183198001838, 7.85239515845810220773035705573, 8.588748999409221035322665954146

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