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Properties

Label 2-792-11.5-c1-0-14
Degree $2$
Conductor $792$
Sign $-0.954 + 0.299i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
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  + (1.02 − 3.14i)5-s + (−3.35 + 2.43i)7-s + (−2.61 + 2.04i)11-s + (−0.752 − 2.31i)13-s + (1.23 − 3.79i)17-s + (−5.57 − 4.04i)19-s + 1.74·23-s + (−4.81 − 3.49i)25-s + (−5.58 + 4.05i)29-s + (−1.82 − 5.61i)31-s + (4.24 + 13.0i)35-s + (−8.68 + 6.30i)37-s + (5.35 + 3.89i)41-s − 3.03·43-s + (−0.750 − 0.544i)47-s + ⋯
L(s)  = 1  + (0.457 − 1.40i)5-s + (−1.26 + 0.922i)7-s + (−0.787 + 0.616i)11-s + (−0.208 − 0.641i)13-s + (0.299 − 0.920i)17-s + (−1.27 − 0.928i)19-s + 0.363·23-s + (−0.962 − 0.698i)25-s + (−1.03 + 0.753i)29-s + (−0.327 − 1.00i)31-s + (0.717 + 2.20i)35-s + (−1.42 + 1.03i)37-s + (0.836 + 0.607i)41-s − 0.463·43-s + (−0.109 − 0.0794i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.954 + 0.299i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0745238 - 0.485939i\)
\(L(\frac12)\) \(\approx\) \(0.0745238 - 0.485939i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (2.61 - 2.04i)T \)
good5 \( 1 + (-1.02 + 3.14i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.35 - 2.43i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.752 + 2.31i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.23 + 3.79i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.57 + 4.04i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 + (5.58 - 4.05i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.82 + 5.61i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.68 - 6.30i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.35 - 3.89i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.03T + 43T^{2} \)
47 \( 1 + (0.750 + 0.544i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.61 + 11.1i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.40 - 1.02i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.26 - 13.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.99T + 67T^{2} \)
71 \( 1 + (-3.56 + 10.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.27 + 1.65i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.31 + 13.2i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.281 + 0.866i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (-1.48 - 4.58i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.636296288508500464323941841676, −9.172278773343649583803684942291, −8.437431220356647500037891300217, −7.31094872708849895937416510651, −6.23147237748637148835205968880, −5.31218471089871329191700634906, −4.76661674091608039463755924597, −3.16309511333389474799607150399, −2.10255392903522667717971839679, −0.22311689370199697604742271026, 2.11488807852265044708655178601, 3.30410329528561816012726492985, 3.93647485862287996400429965618, 5.67645166656663080118708328383, 6.42261788532024267282497490304, 7.01939897572346815176185370974, 7.930065751418102271148502505841, 9.153429811708449762255178351664, 10.08693381797925599218128103374, 10.60598810374718724174085416840

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