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Properties

Label 2-3920-5.3-c0-0-0
Degree $2$
Conductor $3920$
Sign $-0.229 - 0.973i$
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  + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s − 1.00·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + (1 + i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + i·29-s + (0.707 + 0.707i)33-s + (−1 + i)37-s − 1.00i·39-s + (1 + i)43-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s − 1.00·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + (1 + i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + i·29-s + (0.707 + 0.707i)33-s + (−1 + i)37-s − 1.00i·39-s + (1 + i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (2353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.360914172\)
\(L(\frac12)\) \(\approx\) \(1.360914172\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−8.843522838957131731790203604113, −8.198051813973486014735902122482, −7.48573882039519917450708814385, −6.71778996885544165869776609327, −6.01650509900396672636972589621, −4.87564060655328401226414367512, −4.07280437800404577527269942575, −3.42387560408457610987567007966, −2.91633559672192806992462068900, −1.51680575089813441317902430847, 0.72641337591353147999404367569, 2.01329573705601649636190510124, 2.71179751438656202336865655268, 3.87959741459263097284528336055, 4.62473604242831414998589507190, 5.21302924320673264883171717473, 6.67759146117564108056903845575, 7.00315067327827743935720095742, 7.64427606345789594299610872707, 8.567944625962545499314946539046

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