Properties

Label 2-28e2-7.4-c1-0-1
Degree $2$
Conductor $784$
Sign $-0.266 - 0.963i$
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 + 1.73i)3-s + (−2 − 3.46i)5-s + (−0.499 − 0.866i)9-s + 7.99·15-s + (−1 + 1.73i)17-s + (1 + 1.73i)19-s + (4 + 6.92i)23-s + (−5.49 + 9.52i)25-s − 4.00·27-s + 2·29-s + (−2 + 3.46i)31-s + (3 + 5.19i)37-s + 2·41-s − 8·43-s + (−1.99 + 3.46i)45-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + (−0.894 − 1.54i)5-s + (−0.166 − 0.288i)9-s + 2.06·15-s + (−0.242 + 0.420i)17-s + (0.229 + 0.397i)19-s + (0.834 + 1.44i)23-s + (−1.09 + 1.90i)25-s − 0.769·27-s + 0.371·29-s + (−0.359 + 0.622i)31-s + (0.493 + 0.854i)37-s + 0.312·41-s − 1.21·43-s + (−0.298 + 0.516i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433896 + 0.570350i\)
\(L(\frac12)\) \(\approx\) \(0.433896 + 0.570350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.51626774879680361853368289425, −9.670811966317645681185918411209, −8.910297209910759403257150780537, −8.159118395749300509145893752973, −7.23279691262199210163764178735, −5.72014630663450837996309757767, −5.04246744718464189925255466788, −4.34731053421375344325537493103, −3.50346446316351625518264873692, −1.30355356123409053464692514177, 0.42843877870386109503346527148, 2.32847666281513009834910693752, 3.34226234510873272797157465887, 4.55496294889768908866852938258, 6.00757487158782994957037193588, 6.73584687160972200688912488859, 7.24167264687526580812488089890, 7.959165032623729722878548868736, 9.162424098480658488920603158947, 10.39634671333918661667202689782

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