Properties

Label 2-3920-140.27-c0-0-2
Degree $2$
Conductor $3920$
Sign $0.201 - 0.979i$
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  + (0.382 + 0.923i)5-s + i·9-s + (1.30 + 1.30i)13-s + (0.541 − 0.541i)17-s + (−0.707 + 0.707i)25-s − 1.41i·29-s + (−1.41 − 1.41i)37-s + 1.84i·41-s + (−0.923 + 0.382i)45-s + (−1 + i)53-s − 0.765i·61-s + (−0.707 + 1.70i)65-s + (0.541 + 0.541i)73-s − 81-s + (0.707 + 0.292i)85-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)5-s + i·9-s + (1.30 + 1.30i)13-s + (0.541 − 0.541i)17-s + (−0.707 + 0.707i)25-s − 1.41i·29-s + (−1.41 − 1.41i)37-s + 1.84i·41-s + (−0.923 + 0.382i)45-s + (−1 + i)53-s − 0.765i·61-s + (−0.707 + 1.70i)65-s + (0.541 + 0.541i)73-s − 81-s + (0.707 + 0.292i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.201 - 0.979i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.440031393\)
\(L(\frac12)\) \(\approx\) \(1.440031393\)
\(L(1)\) 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 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 \)
good3 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
17 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.765iT - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 0.765T + T^{2} \)
97 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
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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

−8.849022598388048027575710262388, −7.941436424285597334409033549200, −7.38354658090400940862054029242, −6.49924411528547683309363172984, −6.02699415148740151596672616137, −5.10207624455144724473836894013, −4.18426088842779403541829519437, −3.35181076025225690929464859838, −2.36564819505924779190991791175, −1.58313740405398181243232191503, 0.872987649445326700278638572938, 1.71760481812844285265734726862, 3.28656823967365564178585045849, 3.65415354768336566974724650499, 4.85403733472254199829467729945, 5.57109737662062458289751183861, 6.12069088611026646824955277260, 6.93794937963168955786981800321, 7.961664928221561601814705476526, 8.663814728171911188463888244368

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