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Properties

Label 2-28e2-112.27-c1-0-63
Degree $2$
Conductor $784$
Sign $-0.00706 + 0.999i$
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  + (−1.07 + 0.914i)2-s + (1.91 − 1.91i)3-s + (0.326 − 1.97i)4-s + (0.329 − 0.329i)5-s + (−0.313 + 3.81i)6-s + (1.45 + 2.42i)8-s − 4.32i·9-s + (−0.0540 + 0.657i)10-s + (1.41 − 1.41i)11-s + (−3.15 − 4.40i)12-s + (−4.51 − 4.51i)13-s − 1.26i·15-s + (−3.78 − 1.28i)16-s − 5.40i·17-s + (3.95 + 4.66i)18-s + (−1.39 + 1.39i)19-s + ⋯
L(s)  = 1  + (−0.762 + 0.646i)2-s + (1.10 − 1.10i)3-s + (0.163 − 0.986i)4-s + (0.147 − 0.147i)5-s + (−0.128 + 1.55i)6-s + (0.513 + 0.858i)8-s − 1.44i·9-s + (−0.0170 + 0.207i)10-s + (0.427 − 0.427i)11-s + (−0.909 − 1.27i)12-s + (−1.25 − 1.25i)13-s − 0.326i·15-s + (−0.946 − 0.322i)16-s − 1.31i·17-s + (0.932 + 1.10i)18-s + (−0.319 + 0.319i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00706 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00706 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.00706 + 0.999i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.00706 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973814 - 0.980722i\)
\(L(\frac12)\) \(\approx\) \(0.973814 - 0.980722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 - 0.914i)T \)
7 \( 1 \)
good3 \( 1 + (-1.91 + 1.91i)T - 3iT^{2} \)
5 \( 1 + (-0.329 + 0.329i)T - 5iT^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
13 \( 1 + (4.51 + 4.51i)T + 13iT^{2} \)
17 \( 1 + 5.40iT - 17T^{2} \)
19 \( 1 + (1.39 - 1.39i)T - 19iT^{2} \)
23 \( 1 + 1.90T + 23T^{2} \)
29 \( 1 + (1.12 - 1.12i)T - 29iT^{2} \)
31 \( 1 - 0.868T + 31T^{2} \)
37 \( 1 + (-5.78 - 5.78i)T + 37iT^{2} \)
41 \( 1 - 3.24T + 41T^{2} \)
43 \( 1 + (-6.35 + 6.35i)T - 43iT^{2} \)
47 \( 1 - 2.98T + 47T^{2} \)
53 \( 1 + (4.96 + 4.96i)T + 53iT^{2} \)
59 \( 1 + (6.07 + 6.07i)T + 59iT^{2} \)
61 \( 1 + (0.343 + 0.343i)T + 61iT^{2} \)
67 \( 1 + (2.82 + 2.82i)T + 67iT^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 1.23iT - 79T^{2} \)
83 \( 1 + (-9.03 + 9.03i)T - 83iT^{2} \)
89 \( 1 - 2.32T + 89T^{2} \)
97 \( 1 - 4.69iT - 97T^{2} \)
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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

−9.612802859712565451746967400647, −9.152599041520465173289480008355, −8.145605947486268680164306825309, −7.63838362356856625018765328275, −6.99326470563007430510995620880, −5.96282909842426458203489508838, −4.93663193760162556203771329020, −3.10009292499775683979966223377, −2.12129236686692796713711575013, −0.77029022417390977580349467037, 1.99840276227575247685373417736, 2.75890580663259110249360474898, 4.20617131550596213817081333702, 4.31913056618600319671743341529, 6.32665100614882917922061432590, 7.46414693562695748954446532232, 8.234279337839523162818373348598, 9.241399495198070245417923515610, 9.444137260864846441933785751050, 10.30091885993525365738746705719

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