Properties

Label 2-28e2-28.27-c3-0-22
Degree $2$
Conductor $784$
Sign $0.933 - 0.358i$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.30·3-s + 7.29i·5-s − 8.43·9-s − 21.8i·11-s + 30.5i·13-s − 31.4i·15-s − 55.0i·17-s − 58.7·19-s − 79.6i·23-s + 71.7·25-s + 152.·27-s − 116.·29-s + 110.·31-s + 94.0i·33-s − 376.·37-s + ⋯
L(s)  = 1  − 0.829·3-s + 0.652i·5-s − 0.312·9-s − 0.598i·11-s + 0.651i·13-s − 0.541i·15-s − 0.786i·17-s − 0.708·19-s − 0.722i·23-s + 0.574·25-s + 1.08·27-s − 0.744·29-s + 0.637·31-s + 0.496i·33-s − 1.67·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(46.2574\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ 0.933 - 0.358i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.075540153\)
\(L(\frac12)\) \(\approx\) \(1.075540153\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.30T + 27T^{2} \)
5 \( 1 - 7.29iT - 125T^{2} \)
11 \( 1 + 21.8iT - 1.33e3T^{2} \)
13 \( 1 - 30.5iT - 2.19e3T^{2} \)
17 \( 1 + 55.0iT - 4.91e3T^{2} \)
19 \( 1 + 58.7T + 6.85e3T^{2} \)
23 \( 1 + 79.6iT - 1.21e4T^{2} \)
29 \( 1 + 116.T + 2.43e4T^{2} \)
31 \( 1 - 110.T + 2.97e4T^{2} \)
37 \( 1 + 376.T + 5.06e4T^{2} \)
41 \( 1 - 232. iT - 6.89e4T^{2} \)
43 \( 1 + 260. iT - 7.95e4T^{2} \)
47 \( 1 + 36.5T + 1.03e5T^{2} \)
53 \( 1 - 59.1T + 1.48e5T^{2} \)
59 \( 1 - 707.T + 2.05e5T^{2} \)
61 \( 1 - 323. iT - 2.26e5T^{2} \)
67 \( 1 - 239. iT - 3.00e5T^{2} \)
71 \( 1 + 887. iT - 3.57e5T^{2} \)
73 \( 1 - 550. iT - 3.89e5T^{2} \)
79 \( 1 - 1.24e3iT - 4.93e5T^{2} \)
83 \( 1 - 1.39e3T + 5.71e5T^{2} \)
89 \( 1 + 982. iT - 7.04e5T^{2} \)
97 \( 1 - 441. iT - 9.12e5T^{2} \)
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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

−10.24093661567891628976893006491, −9.043806727294480008818042794389, −8.344368616236539004399309084531, −7.02569653486332502212722593176, −6.51828598275922435048589609754, −5.58443035629332967628630002941, −4.68001010865769391089094790398, −3.41215617634945988188027367925, −2.31636908643622150086100714731, −0.62257089253553375019591919349, 0.57081937387088636796051601035, 1.90464019794720350257078467147, 3.42874811250008645993831007537, 4.65611021134235539496419527962, 5.38827916371006621312988611189, 6.18271721392738965931187040872, 7.17156913669638986588618799503, 8.266034814941413741482565586715, 8.904824687472099710151864195629, 10.02212993577254528750183788132

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