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Properties

Label 2-28e2-28.27-c3-0-52
Degree $2$
Conductor $784$
Sign $-0.777 + 0.629i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.777 + 0.629i$
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: \((2,\ 784,\ (\ :3/2),\ -0.777 + 0.629i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.469068268\)
\(L(\frac12)\) \(\approx\) \(1.469068268\)
\(L(\frac{5}{2})\) 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

−9.276926713861159063191890654003, −8.691047175386357965020720994292, −8.119519716372972920277219060554, −7.15466823006263121360818698674, −5.86641379636526987468243157812, −5.15920637432796119934057458535, −3.81285743333443478376388288374, −3.03208260512544900889587244859, −1.75625150676644680620686581339, −0.32697562098164119102857005327, 1.66124819495078821387008650047, 2.79109870572972934514434202660, 3.52761637535954056494624111262, 4.78356932161407848953192493713, 5.86409029924800467393883042479, 7.08697669464621156310381769992, 7.49638666552611531194570489534, 8.656750713531994751909620792881, 9.342636257859053487904748324902, 10.03035605151330128783247944995

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