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Properties

Label 2-3920-20.19-c0-0-0
Degree $2$
Conductor $3920$
Sign $-i$
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  − i·5-s − 9-s + 2i·13-s + 2i·17-s − 25-s − 2·29-s + i·45-s + 2·65-s + 2i·73-s + 81-s + 2·85-s + 2i·97-s + 2·109-s − 2i·117-s + ⋯
L(s)  = 1  − i·5-s − 9-s + 2i·13-s + 2i·17-s − 25-s − 2·29-s + i·45-s + 2·65-s + 2i·73-s + 81-s + 2·85-s + 2i·97-s + 2·109-s − 2i·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7536688459\)
\(L(\frac12)\) \(\approx\) \(0.7536688459\)
\(L(1)\) 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 \( 1 + iT \)
7 \( 1 \)
good3 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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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

−8.839457742917624730236885057138, −8.288877930214538820440036348875, −7.47101949045727111959641763989, −6.46017603534259207123998851816, −5.87953827448520292808163206630, −5.13493417296452068192629447385, −4.13251387569202791887804554781, −3.72727829830711425869347741413, −2.19577350563495235354461058585, −1.52752880183226603916270052959, 0.40325904393571257818145691232, 2.24684707486381082344021250918, 3.06315498666256162455777069828, 3.44228616225946584095229912673, 4.87260657342359490919098493428, 5.60599793423335018263898031548, 6.08527879990415213721264593576, 7.26430085416523957860440244634, 7.53641197172638003296870852757, 8.368707523433423459779826700280

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