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Properties

Label 2-28e2-7.5-c2-0-30
Degree $2$
Conductor $784$
Sign $0.167 + 0.985i$
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  + (−0.274 + 0.158i)3-s + (4.91 + 2.83i)5-s + (−4.44 + 7.70i)9-s + (−7.94 − 13.7i)11-s − 20.1i·13-s − 1.79·15-s + (0.823 − 0.475i)17-s + (−27.5 − 15.9i)19-s + (13 − 22.5i)23-s + (3.60 + 6.23i)25-s − 5.67i·27-s + 27.7·29-s + (−13.1 + 7.57i)31-s + (4.36 + 2.52i)33-s + (16 − 27.7i)37-s + ⋯
L(s)  = 1  + (−0.0915 + 0.0528i)3-s + (0.982 + 0.567i)5-s + (−0.494 + 0.856i)9-s + (−0.722 − 1.25i)11-s − 1.55i·13-s − 0.119·15-s + (0.0484 − 0.0279i)17-s + (−1.45 − 0.838i)19-s + (0.565 − 0.978i)23-s + (0.144 + 0.249i)25-s − 0.210i·27-s + 0.958·29-s + (−0.423 + 0.244i)31-s + (0.132 + 0.0763i)33-s + (0.432 − 0.748i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.167 + 0.985i$
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: \((2,\ 784,\ (\ :1),\ 0.167 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.445877998\)
\(L(\frac12)\) \(\approx\) \(1.445877998\)
\(L(2)\) 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 \( 1 + (0.274 - 0.158i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.91 - 2.83i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (7.94 + 13.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 20.1iT - 169T^{2} \)
17 \( 1 + (-0.823 + 0.475i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (27.5 + 15.9i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-13 + 22.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 27.7T + 841T^{2} \)
31 \( 1 + (13.1 - 7.57i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-16 + 27.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 17.3iT - 1.68e3T^{2} \)
43 \( 1 - 59.2T + 1.84e3T^{2} \)
47 \( 1 + (-66.1 - 38.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (12.8 + 22.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (59.2 - 34.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (46.9 + 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-43.6 - 75.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 16.4T + 5.04e3T^{2} \)
73 \( 1 + (60.9 - 35.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-20.1 + 34.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 71.5iT - 6.88e3T^{2} \)
89 \( 1 + (66.9 + 38.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 128. iT - 9.40e3T^{2} \)
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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

−10.28865482632760682336178353879, −8.902870197520437915064120975600, −8.318415903651236729440920154054, −7.34354754169842174675810289109, −6.05024787822887071689127099938, −5.72577960449088771352345474058, −4.62419736405383343361612313593, −2.89925220731557876694532866165, −2.48130142089343448879201702643, −0.48190756097285037806750624600, 1.47580169081810471888429323384, 2.41378739543940279005700356796, 4.02282309204967086036779685446, 4.91931134437521018473510994679, 5.95500373349445754971064093510, 6.63231918777504716343371589356, 7.68458536161020766698378596240, 8.892265053021802548692609753722, 9.356552918750474470770230433780, 10.10611639775609770524305156689

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