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Properties

Label 1-28e2-784.467-r0-0-0
Degree $1$
Conductor $784$
Sign $-0.712 - 0.701i$
Analytic cond. $3.64088$
Root an. cond. $3.64088$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.563 + 0.826i)3-s + (−0.997 + 0.0747i)5-s + (−0.365 + 0.930i)9-s + (−0.930 + 0.365i)11-s + (−0.781 − 0.623i)13-s + (−0.623 − 0.781i)15-s + (0.733 + 0.680i)17-s + (0.866 − 0.5i)19-s + (−0.733 + 0.680i)23-s + (0.988 − 0.149i)25-s + (−0.974 + 0.222i)27-s + (−0.974 − 0.222i)29-s + (−0.5 + 0.866i)31-s + (−0.826 − 0.563i)33-s + (−0.294 − 0.955i)37-s + ⋯
L(s)  = 1  + (0.563 + 0.826i)3-s + (−0.997 + 0.0747i)5-s + (−0.365 + 0.930i)9-s + (−0.930 + 0.365i)11-s + (−0.781 − 0.623i)13-s + (−0.623 − 0.781i)15-s + (0.733 + 0.680i)17-s + (0.866 − 0.5i)19-s + (−0.733 + 0.680i)23-s + (0.988 − 0.149i)25-s + (−0.974 + 0.222i)27-s + (−0.974 − 0.222i)29-s + (−0.5 + 0.866i)31-s + (−0.826 − 0.563i)33-s + (−0.294 − 0.955i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(3.64088\)
Root analytic conductor: \(3.64088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 784,\ (0:\ ),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04396881084 + 0.1073649515i\)
\(L(\frac12)\) \(\approx\) \(-0.04396881084 + 0.1073649515i\)
\(L(1)\) \(\approx\) \(0.7198368630 + 0.2529582876i\)
\(L(1)\) \(\approx\) \(0.7198368630 + 0.2529582876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.563 + 0.826i)T \)
5 \( 1 + (-0.997 + 0.0747i)T \)
11 \( 1 + (-0.930 + 0.365i)T \)
13 \( 1 + (-0.781 - 0.623i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (-0.974 - 0.222i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.294 - 0.955i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (0.294 - 0.955i)T \)
59 \( 1 + (-0.997 - 0.0747i)T \)
61 \( 1 + (-0.294 - 0.955i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.781 + 0.623i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.85521135792134501238890086389, −20.59761451891263659176706523899, −20.311377567282650761414744765168, −19.31671514028746279969743717471, −18.56803840862359110133885871479, −18.287317221805058484509376945707, −16.80561169940683120859576355112, −16.23161128359388042208837322597, −15.16471549594462813634056560108, −14.49125387205617731724734053190, −13.65149680488517129361741041497, −12.76553868533700831054927196716, −11.954787914884871652814723659532, −11.449219776046498133190803120602, −10.10182182194684431598131214544, −9.19563580523939894783189821767, −8.08996098489444491699218864665, −7.692549221501917014817770183456, −6.89747494710709588725487026070, −5.69773985805813405355481387161, −4.61233595715904599737063589674, −3.426020079728163993613234469004, −2.72897688973903217596339571009, −1.461595461935282680154376436, −0.04805819265152965332572781135, 1.99998237155573876190920724000, 3.19681957177215367256252999873, 3.712015903284607083545793484719, 4.94097990378187072768591690614, 5.46226412242066864803449656841, 7.254435323860512513458490026693, 7.77093808862214891406809141537, 8.55333767223627454993202908268, 9.67984275402704688989318163455, 10.33193230887217429152876056925, 11.16501629592272609335430272, 12.15016651229271747573121968439, 12.988888155283981781324132518137, 14.07039623479095038628737713585, 14.92226441704139880867597609779, 15.485854340658299434968125125067, 16.09668418223227080398581018745, 17.0109309581460036845995221218, 18.06579814088269782645003254354, 19.01339908737111250450497201731, 19.85061544728433226269968907247, 20.241956741086878566144330369191, 21.18539259429469706141036734463, 22.00458329106545796457002965311, 22.72715180807923228393993423752

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