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Properties

Label 2-28e2-7.5-c2-0-14
Degree $2$
Conductor $784$
Sign $0.832 - 0.553i$
Analytic cond. $21.3624$
Root an. cond. $4.62195$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.20 + 1.84i)3-s + (−0.549 − 0.317i)5-s + (2.32 − 4.03i)9-s + (6.65 + 11.5i)11-s − 21.5i·13-s + 2.34·15-s + (5.30 − 3.06i)17-s + (−17.1 − 9.87i)19-s + (−14.3 + 24.7i)23-s + (−12.2 − 21.3i)25-s − 16.0i·27-s + 37.3·29-s + (18.1 − 10.4i)31-s + (−42.6 − 24.6i)33-s + (11.9 − 20.7i)37-s + ⋯
L(s)  = 1  + (−1.06 + 0.615i)3-s + (−0.109 − 0.0634i)5-s + (0.258 − 0.448i)9-s + (0.605 + 1.04i)11-s − 1.65i·13-s + 0.156·15-s + (0.311 − 0.180i)17-s + (−0.900 − 0.519i)19-s + (−0.622 + 1.07i)23-s + (−0.491 − 0.852i)25-s − 0.594i·27-s + 1.28·29-s + (0.584 − 0.337i)31-s + (−1.29 − 0.745i)33-s + (0.323 − 0.560i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(21.3624\)
Root analytic conductor: \(4.62195\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (705, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.098406475\)
\(L(\frac12)\) \(\approx\) \(1.098406475\)
\(L(2)\) 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 \( 1 + (3.20 - 1.84i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (0.549 + 0.317i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.65 - 11.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.5iT - 169T^{2} \)
17 \( 1 + (-5.30 + 3.06i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (17.1 + 9.87i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (14.3 - 24.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 37.3T + 841T^{2} \)
31 \( 1 + (-18.1 + 10.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.9 + 20.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 70.1iT - 1.68e3T^{2} \)
43 \( 1 - 46T + 1.84e3T^{2} \)
47 \( 1 + (0.909 + 0.525i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-7.68 - 13.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-12.8 + 7.44i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-75.1 - 43.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-12.3 - 21.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 45.0T + 5.04e3T^{2} \)
73 \( 1 + (44.8 - 25.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (16.8 - 29.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 77.3iT - 6.88e3T^{2} \)
89 \( 1 + (-68.2 - 39.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 90.1iT - 9.40e3T^{2} \)
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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

−10.08509905875817612860904386142, −9.783646862939400548883798033795, −8.404631770077347433267400206574, −7.61498430779994358844063516759, −6.44121971280626354537378495314, −5.70537469577929694670681403965, −4.78101146252588726167985003780, −4.04332430712252488749810298671, −2.55332649753593173583682657237, −0.74890658811755181233828003111, 0.70539796987481426467742208525, 1.97083532282114661805479504145, 3.64725146546249918614892289639, 4.63289948449977097285813507867, 5.94103160558674009128256159829, 6.37268474997228053298153693827, 7.12642005715236925630560734535, 8.372963241963310365134730375839, 9.033399162339304339520424485692, 10.23239047461490852309642125904

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