L-function 1-28e2-784.467-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	7	$1$
good	3	$1 + (0.563 + 0.826i)T$
	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	$1 + (-0.5 + 0.866i)T$

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