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Properties

Label 2-28e2-28.27-c1-0-9
Degree $2$
Conductor $784$
Sign $0.755 - 0.654i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
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  + 2.64·3-s + 1.73i·5-s + 4.00·9-s + 4.58i·11-s − 3.46i·13-s + 4.58i·15-s + 5.19i·17-s + 2.64·19-s − 4.58i·23-s + 2.00·25-s + 2.64·27-s − 2.64·31-s + 12.1i·33-s + 7·37-s − 9.16i·39-s + ⋯
L(s)  = 1  + 1.52·3-s + 0.774i·5-s + 1.33·9-s + 1.38i·11-s − 0.960i·13-s + 1.18i·15-s + 1.26i·17-s + 0.606·19-s − 0.955i·23-s + 0.400·25-s + 0.509·27-s − 0.475·31-s + 2.11i·33-s + 1.15·37-s − 1.46i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39893 + 0.894383i\)
\(L(\frac12)\) \(\approx\) \(2.39893 + 0.894383i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 - 4.58iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 5.19iT - 17T^{2} \)
19 \( 1 - 2.64T + 19T^{2} \)
23 \( 1 + 4.58iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 9.16iT - 43T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 + 1.73iT - 61T^{2} \)
67 \( 1 + 4.58iT - 67T^{2} \)
71 \( 1 + 9.16iT - 71T^{2} \)
73 \( 1 + 5.19iT - 73T^{2} \)
79 \( 1 + 4.58iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 1.73iT - 89T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−10.21666436426837769871106144216, −9.563882351279743273455760844042, −8.610558489760635204837870951647, −7.83821062609616349995624810913, −7.22650335556974889495287054308, −6.20678833967061048478611712486, −4.75585728871569662092256299835, −3.66027575408347676145119969676, −2.82784359114572337154693213946, −1.87264022378996563035479341153, 1.24265905466638619844136514880, 2.68400497140023709354755558156, 3.51614842757731313082074368696, 4.59423833393748480326918488903, 5.68195131611351727254162070929, 6.99338158235620623193801213627, 7.87274332310234404532035344899, 8.587448687362488269378522988300, 9.278305293491290063983493604755, 9.663860725406179085747627895606

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