L-function 1-520-520.493-r1-0-0

• Label: 1-520-520.493-r1-0-0
• Degree: $1$
• Conductor: $520$
• Sign: $-0.850 + 0.525i$
• Analytic cond.: $55.8817$
• Root an. cond.: $55.8817$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− i·3^-s + i·7^-s − 9^-s + 11^-s − i·17^-s − 19^-s + 21^-s + i·23^-s + i·27^-s + 29^-s − 31^-s − i·33^-s − i·37^-s − 41^-s − i·43^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign:	$-0.850 + 0.525i$
Analytic conductor:	\(55.8817\)
Root analytic conductor:	\(55.8817\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{520} (493, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 520,\ (1:\ ),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.01930139266 - 0.06794373990i\)
\(L(1)\)	\(\approx\)	\(0.8480123126 - 0.2001885515i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	5	\( 1 \)
	13	\( 1 \)
good	3	\( 1 + T \)
	7	\( 1 - iT \)
	11	\( 1 \)
	17	\( 1 \)
	19	\( 1 + iT \)
	23	\( 1 \)
	29	\( 1 - T \)
	31	\( 1 \)
	37	\( 1 + T \)

Imaginary part of the first few zeros on the critical line

−23.56218803276308000524481410951, −23.00492538527100197324209269156, −21.971086305800787782571113277775, −21.464765442764555908696234518357, −20.31840612311396798709513458692, −19.923128081188688595991809990167, −18.946172486106917322991229230647, −17.53810431449030372007993160239, −16.90591125439961488089495610285, −16.394746397736307090195233211229, −15.14600155069740803304946760773, −14.57410951031407983921122063622, −13.7275296936409919341543304530, −12.59333644712353988160682835280, −11.5099390177137716255848484271, −10.58436502027786263374803948811, −10.09418311986337203823996064975, −8.92473220832702080804350338515, −8.216172161163277057462121740285, −6.81439233338221299678401321338, −6.026551160440865668399088586225, −4.57899389776792031405443250448, −4.09665955017170492073809404139, −3.06143435230847750324050643443, −1.48378498217064249556033326121, 0.01735230950147015540040240873, 1.45633552525518536843757208962, 2.3713881735861690967443408574, 3.48120666243944729733399832592, 4.98951631022137095977471961425, 5.9869815085371620619738077486, 6.751515667655515034639847054659, 7.730657252379397089936852931009, 8.80725972413671467002348224740, 9.34359653656719836718119895229, 10.89425172352993276556532382311, 11.84621675824583380136748736478, 12.29653052955928705646995156961, 13.33045800773207979073688812662, 14.20847034031471918753003124793, 14.98107530401516279808912800761, 16.044153620186619589674954899086, 17.10524271783263660829155274176, 17.84489551938149370332448568582, 18.6404338334639499412801417873, 19.37414921225198071260376136809, 20.05285991863407142204632498253, 21.23832838138698756719441007555, 22.08991291743246108139949298546, 22.88921246451969237927829954438

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