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Properties

Label 2-792-24.11-c1-0-7
Degree $2$
Conductor $792$
Sign $0.869 - 0.493i$
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  + (0.747 − 1.20i)2-s + (−0.882 − 1.79i)4-s − 1.25·5-s + 4.96i·7-s + (−2.81 − 0.281i)8-s + (−0.941 + 1.51i)10-s − i·11-s + 1.75i·13-s + (5.96 + 3.71i)14-s + (−2.44 + 3.16i)16-s + 5.31i·17-s + 5.49·19-s + (1.11 + 2.25i)20-s + (−1.20 − 0.747i)22-s + 0.569·23-s + ⋯
L(s)  = 1  + (0.528 − 0.848i)2-s + (−0.441 − 0.897i)4-s − 0.563·5-s + 1.87i·7-s + (−0.995 − 0.0994i)8-s + (−0.297 + 0.478i)10-s − 0.301i·11-s + 0.485i·13-s + (1.59 + 0.992i)14-s + (−0.610 + 0.792i)16-s + 1.28i·17-s + 1.26·19-s + (0.248 + 0.505i)20-s + (−0.255 − 0.159i)22-s + 0.118·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33788 + 0.352954i\)
\(L(\frac12)\) \(\approx\) \(1.33788 + 0.352954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.747 + 1.20i)T \)
3 \( 1 \)
11 \( 1 + iT \)
good5 \( 1 + 1.25T + 5T^{2} \)
7 \( 1 - 4.96iT - 7T^{2} \)
13 \( 1 - 1.75iT - 13T^{2} \)
17 \( 1 - 5.31iT - 17T^{2} \)
19 \( 1 - 5.49T + 19T^{2} \)
23 \( 1 - 0.569T + 23T^{2} \)
29 \( 1 - 2.14T + 29T^{2} \)
31 \( 1 - 6.43iT - 31T^{2} \)
37 \( 1 - 6.33iT - 37T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + 0.0868T + 43T^{2} \)
47 \( 1 - 2.17T + 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 + 15.1iT - 59T^{2} \)
61 \( 1 - 8.15iT - 61T^{2} \)
67 \( 1 + 3.41T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 9.33T + 73T^{2} \)
79 \( 1 + 12.8iT - 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 - 7.79iT - 89T^{2} \)
97 \( 1 + 5.97T + 97T^{2} \)
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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

−10.49667671685256410059696639784, −9.511280124176535987521640777471, −8.812144073307112220901039393524, −8.101109041989022575884063633960, −6.54714119466601243118284539861, −5.70451633458220284778203352175, −4.97684766455163468718061255429, −3.72692865301623495592092370888, −2.82557126199069557312928722526, −1.66160214814526680770693176549, 0.60262518391625913942254344529, 3.06271899812404792657971878786, 4.00435745723603034706195404806, 4.68759249200171022464231005186, 5.78013243893447043358473647712, 7.09287604653471711690951654628, 7.39800233421042074439115759561, 8.038544970866169766342513194412, 9.364097947234605651999039360626, 10.08527436845836950063174581072

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