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Properties

Label 2-828-12.11-c1-0-3
Degree $2$
Conductor $828$
Sign $-0.735 - 0.677i$
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  + (−1.06 + 0.933i)2-s + (0.257 − 1.98i)4-s + 1.22i·5-s − 3.40i·7-s + (1.57 + 2.34i)8-s + (−1.14 − 1.30i)10-s − 2.21·11-s − 4.82·13-s + (3.18 + 3.62i)14-s + (−3.86 − 1.02i)16-s + 1.01i·17-s + 8.63i·19-s + (2.43 + 0.316i)20-s + (2.35 − 2.07i)22-s + 23-s + ⋯
L(s)  = 1  + (−0.751 + 0.660i)2-s + (0.128 − 0.991i)4-s + 0.549i·5-s − 1.28i·7-s + (0.557 + 0.830i)8-s + (−0.362 − 0.412i)10-s − 0.668·11-s − 1.33·13-s + (0.850 + 0.967i)14-s + (−0.966 − 0.255i)16-s + 0.245i·17-s + 1.98i·19-s + (0.544 + 0.0707i)20-s + (0.502 − 0.441i)22-s + 0.208·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ -0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211393 + 0.541306i\)
\(L(\frac12)\) \(\approx\) \(0.211393 + 0.541306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 - 0.933i)T \)
3 \( 1 \)
23 \( 1 - T \)
good5 \( 1 - 1.22iT - 5T^{2} \)
7 \( 1 + 3.40iT - 7T^{2} \)
11 \( 1 + 2.21T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 1.01iT - 17T^{2} \)
19 \( 1 - 8.63iT - 19T^{2} \)
29 \( 1 - 9.68iT - 29T^{2} \)
31 \( 1 - 3.37iT - 31T^{2} \)
37 \( 1 - 4.73T + 37T^{2} \)
41 \( 1 - 3.51iT - 41T^{2} \)
43 \( 1 + 3.87iT - 43T^{2} \)
47 \( 1 + 2.64T + 47T^{2} \)
53 \( 1 - 13.5iT - 53T^{2} \)
59 \( 1 - 0.774T + 59T^{2} \)
61 \( 1 + 9.56T + 61T^{2} \)
67 \( 1 - 8.78iT - 67T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 8.25iT - 79T^{2} \)
83 \( 1 + 2.16T + 83T^{2} \)
89 \( 1 + 4.80iT - 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−10.46932711890677886294401778572, −9.872076714226325263957760544471, −8.772586522560943536018278879520, −7.66026689109076208528474156477, −7.38389761509776466414727780063, −6.47973153914880562600809772070, −5.39996061699711576735746646655, −4.41633546564410314246725772343, −2.99437205571695855210351541060, −1.42570381839805746995489863261, 0.36941585282810674337562493843, 2.31470276681472745696770272810, 2.78249844984343668347069390653, 4.53837589757058442265486428203, 5.23535092141870778706417103317, 6.60140599616383750732503670391, 7.63783254534029244453987903458, 8.368222312693917238953911705346, 9.341197229073572844616286958672, 9.538149815657038200733937944623

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