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Properties

Label 2-828-23.2-c1-0-8
Degree $2$
Conductor $828$
Sign $0.778 + 0.628i$
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 + 0.701i)5-s + (2.64 − 3.05i)7-s + (4.05 − 2.60i)11-s + (−2.81 − 3.24i)13-s + (−1.87 + 4.10i)17-s + (−1.84 − 4.02i)19-s + (−4.75 + 0.655i)23-s + (1.01 + 0.649i)25-s + (0.207 − 0.454i)29-s + (−0.727 + 5.06i)31-s + (8.47 − 5.44i)35-s + (10.2 − 3.00i)37-s + (−7.29 − 2.14i)41-s + (1.07 + 7.50i)43-s + 7.67·47-s + ⋯
L(s)  = 1  + (1.06 + 0.313i)5-s + (1.00 − 1.15i)7-s + (1.22 − 0.786i)11-s + (−0.780 − 0.900i)13-s + (−0.454 + 0.995i)17-s + (−0.422 − 0.924i)19-s + (−0.990 + 0.136i)23-s + (0.202 + 0.129i)25-s + (0.0385 − 0.0843i)29-s + (−0.130 + 0.909i)31-s + (1.43 − 0.920i)35-s + (1.68 − 0.494i)37-s + (−1.13 − 0.334i)41-s + (0.164 + 1.14i)43-s + 1.11·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.778 + 0.628i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ 0.778 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93870 - 0.684966i\)
\(L(\frac12)\) \(\approx\) \(1.93870 - 0.684966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.75 - 0.655i)T \)
good5 \( 1 + (-2.38 - 0.701i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (-2.64 + 3.05i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (-4.05 + 2.60i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.81 + 3.24i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (1.87 - 4.10i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (1.84 + 4.02i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-0.207 + 0.454i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.727 - 5.06i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (-10.2 + 3.00i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (7.29 + 2.14i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-1.07 - 7.50i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 7.67T + 47T^{2} \)
53 \( 1 + (5.00 - 5.77i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-1.85 - 2.13i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.225 - 1.57i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-10.1 - 6.49i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (3.18 + 2.04i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (3.09 + 6.77i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-7.62 - 8.79i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-5.09 + 1.49i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.58 - 11.0i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (6.84 + 2.00i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.26938334325532167485961875750, −9.371529796139415558737859933277, −8.415279963821225924703026816855, −7.57451811067432619098146006326, −6.56576200508991872319650096273, −5.86857886068102884851000832324, −4.69786191659177068328165983041, −3.79010878088386025473235986462, −2.33664498452047723703700202526, −1.12325269516250205545818052176, 1.81683850779930164422639325079, 2.22564651652538786588617307021, 4.19704648459817432755822620398, 4.99073543425566634870992171985, 5.90300876501399832075732219026, 6.70582874050847161673361397936, 7.84645472274906556917344271898, 8.843815162404991666812286294690, 9.469562367349605133098918120129, 9.966197770405078156106728314828

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