Properties

Label 2-28e2-28.27-c3-0-21
Degree $2$
Conductor $784$
Sign $0.944 + 0.327i$
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  − 4.76·3-s − 16.8i·5-s − 4.26·9-s − 40.7i·11-s + 56.7i·13-s + 80.5i·15-s + 122. i·17-s + 75.4·19-s + 106. i·23-s − 160.·25-s + 149.·27-s − 146.·29-s + 42.5·31-s + 194. i·33-s + 80.9·37-s + ⋯
L(s)  = 1  − 0.917·3-s − 1.51i·5-s − 0.157·9-s − 1.11i·11-s + 1.21i·13-s + 1.38i·15-s + 1.75i·17-s + 0.911·19-s + 0.962i·23-s − 1.28·25-s + 1.06·27-s − 0.939·29-s + 0.246·31-s + 1.02i·33-s + 0.359·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.944 + 0.327i$
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: \((2,\ 784,\ (\ :3/2),\ 0.944 + 0.327i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.146900145\)
\(L(\frac12)\) \(\approx\) \(1.146900145\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.76T + 27T^{2} \)
5 \( 1 + 16.8iT - 125T^{2} \)
11 \( 1 + 40.7iT - 1.33e3T^{2} \)
13 \( 1 - 56.7iT - 2.19e3T^{2} \)
17 \( 1 - 122. iT - 4.91e3T^{2} \)
19 \( 1 - 75.4T + 6.85e3T^{2} \)
23 \( 1 - 106. iT - 1.21e4T^{2} \)
29 \( 1 + 146.T + 2.43e4T^{2} \)
31 \( 1 - 42.5T + 2.97e4T^{2} \)
37 \( 1 - 80.9T + 5.06e4T^{2} \)
41 \( 1 + 53.8iT - 6.89e4T^{2} \)
43 \( 1 + 341. iT - 7.95e4T^{2} \)
47 \( 1 - 4.12T + 1.03e5T^{2} \)
53 \( 1 - 279.T + 1.48e5T^{2} \)
59 \( 1 - 174.T + 2.05e5T^{2} \)
61 \( 1 - 467. iT - 2.26e5T^{2} \)
67 \( 1 - 753. iT - 3.00e5T^{2} \)
71 \( 1 + 669. iT - 3.57e5T^{2} \)
73 \( 1 - 835. iT - 3.89e5T^{2} \)
79 \( 1 + 1.10e3iT - 4.93e5T^{2} \)
83 \( 1 + 552.T + 5.71e5T^{2} \)
89 \( 1 + 122. iT - 7.04e5T^{2} \)
97 \( 1 + 291. iT - 9.12e5T^{2} \)
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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

−9.812773627264119850130483960969, −8.798827801516222662985172490386, −8.483300684942479815770468165659, −7.21840258970836928054225102158, −5.83648141245642652256348070866, −5.69334251487815039721661505672, −4.54770489748745336435564812274, −3.61186601940660291716316729121, −1.69222448816147821675605662581, −0.70327154519702883421232266857, 0.56960358216207807276808639931, 2.49496298323615690000981164823, 3.20994138113759557299512675351, 4.75018006116760922871491012048, 5.55374187807768497615064258687, 6.52363998027358737059825638543, 7.19361974036652705197552812861, 7.914459096100567625949019035403, 9.438895508779303440596602162922, 10.10431420760663204929793015363

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