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Properties

Label 2-3192-3192.797-c0-0-20
Degree $2$
Conductor $3192$
Sign $1$
Analytic cond. $1.59301$
Root an. cond. $1.26214$
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  + 2-s − i·3-s + 4-s + 2i·5-s − i·6-s + 7-s + 8-s − 9-s + 2i·10-s − i·12-s − 2i·13-s + 14-s + 2·15-s + 16-s − 18-s + i·19-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s + 2i·5-s − i·6-s + 7-s + 8-s − 9-s + 2i·10-s − i·12-s − 2i·13-s + 14-s + 2·15-s + 16-s − 18-s + i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(1.59301\)
Root analytic conductor: \(1.26214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3192} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3192,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.626903433\)
\(L(\frac12)\) \(\approx\) \(2.626903433\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + iT \)
7 \( 1 - T \)
19 \( 1 - iT \)
good5 \( 1 - 2iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
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 \( 1 + 2T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 - T^{2} \)
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

−8.268615859376721238835279066319, −7.77890167391301296388048214636, −7.33282346279644752314708031636, −6.55370510825433869526027986228, −5.80996117496152935740645317042, −5.39344144220987535587836465336, −3.93919233747926946379096691669, −3.06905352953739779879424344563, −2.58487375487441571902150229378, −1.57876075143363064038193750824, 1.40277724394777344256169987574, 2.29589231477238962021497273116, 3.82072583867059722238170886956, 4.50017289460094680775993294829, 4.72465966697653627683405540611, 5.40744082925062587124427262978, 6.24256198640426821314259863087, 7.38131304902125344300158087262, 8.264928951684689707314588135520, 8.967346389423622737468225852640

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