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Properties

Label 2-828-23.3-c1-0-8
Degree $2$
Conductor $828$
Sign $-0.464 + 0.885i$
Analytic cond. $6.61161$
Root an. cond. $2.57130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.38 − 2.74i)5-s + (−3.67 + 2.36i)7-s + (0.389 − 2.70i)11-s + (−5.17 − 3.32i)13-s + (−1.26 − 0.370i)17-s + (5.16 − 1.51i)19-s + (−0.596 − 4.75i)23-s + (−1.17 − 8.14i)25-s + (−6.27 − 1.84i)29-s + (1.69 + 3.72i)31-s + (−2.26 + 15.7i)35-s + (−3.77 − 4.35i)37-s + (5.09 − 5.87i)41-s + (−3.90 + 8.56i)43-s + 5.07·47-s + ⋯
L(s)  = 1  + (1.06 − 1.22i)5-s + (−1.38 + 0.892i)7-s + (0.117 − 0.816i)11-s + (−1.43 − 0.922i)13-s + (−0.306 − 0.0898i)17-s + (1.18 − 0.348i)19-s + (−0.124 − 0.992i)23-s + (−0.234 − 1.62i)25-s + (−1.16 − 0.341i)29-s + (0.305 + 0.668i)31-s + (−0.382 + 2.65i)35-s + (−0.620 − 0.716i)37-s + (0.795 − 0.917i)41-s + (−0.596 + 1.30i)43-s + 0.740·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.464 + 0.885i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ -0.464 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580495 - 0.960205i\)
\(L(\frac12)\) \(\approx\) \(0.580495 - 0.960205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 + (0.596 + 4.75i)T \)
good5 \( 1 + (-2.38 + 2.74i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (3.67 - 2.36i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.389 + 2.70i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (5.17 + 3.32i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (1.26 + 0.370i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-5.16 + 1.51i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (6.27 + 1.84i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-1.69 - 3.72i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (3.77 + 4.35i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-5.09 + 5.87i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.90 - 8.56i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 5.07T + 47T^{2} \)
53 \( 1 + (3.43 - 2.20i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-2.90 - 1.86i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (1.10 + 2.41i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.627 - 4.36i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (1.45 + 10.1i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-6.23 + 1.83i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (2.84 + 1.82i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (4.71 + 5.43i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-4.07 + 8.92i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (1.60 - 1.85i)T + (-13.8 - 96.0i)T^{2} \)
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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

−9.645536247278961379366685114633, −9.272536991979274437535968313289, −8.559022478014071877546277923946, −7.34884082961067048164553229473, −6.14207520042251826077585027267, −5.59600429827712807853879135989, −4.83994032445354376542867190979, −3.20133631184883582277424535598, −2.29734055346497104043029305388, −0.51420838811021627803095037409, 1.92801504457033699107654550553, 2.98220108341934835136729510183, 3.95999199255804631920645870171, 5.33839700336335683462407218855, 6.37347899086939399525106479062, 7.09903572666976808837292641227, 7.40102723974424771792739029110, 9.377490217561764759485246681732, 9.775545581865955513165017541344, 10.11536249439404668676149500286

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