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Properties

Label 2-3920-35.2-c0-0-2
Degree $2$
Conductor $3920$
Sign $0.164 + 0.986i$
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  + (−0.965 − 0.258i)3-s + (−0.258 − 0.965i)5-s + (0.5 + 0.866i)11-s + (0.707 − 0.707i)13-s + i·15-s + (0.258 − 0.965i)17-s + (1.22 + 0.707i)19-s + (−0.866 + 0.499i)25-s + (0.707 + 0.707i)27-s + i·29-s + (−0.707 − 1.22i)31-s + (−0.258 − 0.965i)33-s + (−0.866 + 0.500i)39-s + 1.41·41-s + (1 − i)43-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (−0.258 − 0.965i)5-s + (0.5 + 0.866i)11-s + (0.707 − 0.707i)13-s + i·15-s + (0.258 − 0.965i)17-s + (1.22 + 0.707i)19-s + (−0.866 + 0.499i)25-s + (0.707 + 0.707i)27-s + i·29-s + (−0.707 − 1.22i)31-s + (−0.258 − 0.965i)33-s + (−0.866 + 0.500i)39-s + 1.41·41-s + (1 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.164 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8745458642\)
\(L(\frac12)\) \(\approx\) \(0.8745458642\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 \)
good3 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.506349220047682887580294724556, −7.52385118681484880276230669601, −7.20122800462267180190911471359, −5.93648090362055887279149879915, −5.64242096375733336076448900567, −4.86796126863243755471787909010, −4.02661618366821646763361621364, −3.07876189839319018345468376679, −1.58312485325401905286548299135, −0.70888816020415447372808097668, 1.11670011285545994074245164372, 2.57007833913828913835473860707, 3.51244481412788289076372996183, 4.16323661082530831659113954793, 5.22478320209833644844067217562, 6.02668395681334176030986786261, 6.36496533341369222759944886654, 7.21196969732396091791994502594, 8.034115103440020282004580283702, 8.811511591113802887649920303929

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