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Properties

Label 2-3920-980.739-c0-0-0
Degree $2$
Conductor $3920$
Sign $0.0106 + 0.999i$
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  + (−0.432 − 0.400i)3-s + (−0.955 + 0.294i)5-s + (0.930 − 0.365i)7-s + (−0.0487 − 0.650i)9-s + (0.531 + 0.255i)15-s + (−0.548 − 0.215i)21-s + (1.34 + 0.202i)23-s + (0.826 − 0.563i)25-s + (−0.607 + 0.761i)27-s + (−1.23 − 1.54i)29-s + (−0.781 + 0.623i)35-s + (0.0332 − 0.145i)41-s + (0.443 + 1.94i)43-s + (0.238 + 0.607i)45-s + (−1.61 − 1.09i)47-s + ⋯
L(s)  = 1  + (−0.432 − 0.400i)3-s + (−0.955 + 0.294i)5-s + (0.930 − 0.365i)7-s + (−0.0487 − 0.650i)9-s + (0.531 + 0.255i)15-s + (−0.548 − 0.215i)21-s + (1.34 + 0.202i)23-s + (0.826 − 0.563i)25-s + (−0.607 + 0.761i)27-s + (−1.23 − 1.54i)29-s + (−0.781 + 0.623i)35-s + (0.0332 − 0.145i)41-s + (0.443 + 1.94i)43-s + (0.238 + 0.607i)45-s + (−1.61 − 1.09i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.0106 + 0.999i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.0106 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8960775242\)
\(L(\frac12)\) \(\approx\) \(0.8960775242\)
\(L(1)\) 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 \( 1 + (0.955 - 0.294i)T \)
7 \( 1 + (-0.930 + 0.365i)T \)
good3 \( 1 + (0.432 + 0.400i)T + (0.0747 + 0.997i)T^{2} \)
11 \( 1 + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.34 - 0.202i)T + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (1.23 + 1.54i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (-0.0332 + 0.145i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (-0.443 - 1.94i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (1.61 + 1.09i)T + (0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (-0.535 + 1.36i)T + (-0.733 - 0.680i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.67 - 0.807i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2} \)
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

−8.158413462099373611988671246187, −7.79362205658283889779202356411, −6.98493045526776580449108510931, −6.45544113273780563406048125688, −5.45018422825424319766677793051, −4.66615936847665238303235222493, −3.87005482369146278291778634836, −3.09640343441813640924207174639, −1.76823912569376666878658058286, −0.60163448266495867428758684935, 1.27341048470261553855634000831, 2.46778609303364155900558912195, 3.59376689497244035068065858230, 4.40030487190126310685211051283, 5.21141789892260570590701275451, 5.39148391448491280462120292442, 6.77352177533520587839004517360, 7.46611401963742211041812082361, 8.095629919904190113927418329656, 8.780441874668275380271948830659

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