L-function 1-260-260.63-r1-0-0

• Label: 1-260-260.63-r1-0-0
• Degree: $1$
• Conductor: $260$
• Sign: $0.507 - 0.861i$
• Analytic cond.: $27.9408$
• Root an. cond.: $27.9408$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.866 + 0.5i)3^-s + (−0.5 − 0.866i)7^-s + (0.5 + 0.866i)9^-s + (−0.866 − 0.5i)11^-s + (0.866 − 0.5i)17^-s + (0.866 − 0.5i)19^-s − i·21^-s + (−0.866 − 0.5i)23^-s − i·27^-s + (0.5 − 0.866i)29^-s − i·31^-s + (−0.5 − 0.866i)33^-s + (0.5 − 0.866i)37^-s + (−0.866 − 0.5i)41^-s + (0.866 − 0.5i)43^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$260$    =    $2^{2} \cdot 5 \cdot 13$
Sign:	$0.507 - 0.861i$
Analytic conductor:	$27.9408$
Root analytic conductor:	$27.9408$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{260} (63, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 260,\ (1:\ ),\ 0.507 - 0.861i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.822758708 - 1.041899411i$
$L(1)$	$\approx$	$1.312810804 - 0.1312926369i$

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	$1 + (0.866 + 0.5i)T$
	7	$1 + (-0.5 - 0.866i)T$
	11	$1 + (-0.866 - 0.5i)T$
	17	$1 + (0.866 - 0.5i)T$
	19	$1 + (0.866 - 0.5i)T$
	23	$1 + (-0.866 - 0.5i)T$
	29	$1 + (0.5 - 0.866i)T$
	31	$1 - iT$
	37	$1 + (0.5 - 0.866i)T$

Imaginary part of the first few zeros on the critical line

−25.583928323700287961963532751816, −25.206168569763524261480117580739, −24.02266122005873726904885112613, −23.290965747288920591652626400624, −22.04017089853320294222504821668, −21.14421100312114675589791487467, −20.23167592092980421483689462168, −19.34774583802021232037598951193, −18.457631948575509011702718271188, −17.88506912559648435351076964497, −16.27540252963815161535014430299, −15.464248701606341753341851827003, −14.56793760519405529815636741828, −13.58473234802728342005203243606, −12.553846007747722091441037467961, −12.013548941639456545658188983372, −10.25274582633549496868766473834, −9.47159333224783574392736922190, −8.34278082406918272749321932905, −7.56515215377453961376923581436, −6.341908378475445631591330873497, −5.20230907074784218563422733588, −3.53886124626226544077745287818, −2.65682201853045912544269988971, −1.42268606674504629986636616148, 0.60087705055952565328322565218, 2.46719053514251435929966550265, 3.44140664429480418492133401363, 4.483167533990135843535419873102, 5.77821712462214185087862184479, 7.35156922749827283992258296057, 7.99407289381297991648867681743, 9.31318671453022153201598954318, 10.09773417040597860770460700691, 10.94723610940759706655452771435, 12.42370290163529432160450438350, 13.72479566428842979748464616618, 13.90547121117239172820511183602, 15.36634406328652819781028537285, 16.08334618624184817863400168235, 16.901814151686133411270926231353, 18.3456502801372696445070517136, 19.16698664354184127718295338909, 20.18806815173830763044901536728, 20.72746334619556489974935915217, 21.75524911904627252097887331839, 22.7074999645223900067723451944, 23.74692921812292220147180102911, 24.68783112653183236055526368392, 25.75854807289780962058229277133

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