L-function 1-260-260.43-r0-0-0

• Label: 1-260-260.43-r0-0-0
• Degree: $1$
• Conductor: $260$
• Sign: $0.874 + 0.485i$
• Analytic cond.: $1.20743$
• Root an. cond.: $1.20743$
• 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 + (0.866 + 0.5i)7^-s + (0.5 − 0.866i)9^-s + (−0.5 − 0.866i)11^-s + (0.866 + 0.5i)17^-s + (0.5 − 0.866i)19^-s − 21^-s + (−0.866 + 0.5i)23^-s − i·27^-s + (0.5 + 0.866i)29^-s + 31^-s + (0.866 + 0.5i)33^-s + (0.866 − 0.5i)37^-s + (0.5 + 0.866i)41^-s + (0.866 + 0.5i)43^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$260$    =    $2^{2} \cdot 5 \cdot 13$
Sign:	$0.874 + 0.485i$
Analytic conductor:	$1.20743$
Root analytic conductor:	$1.20743$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{260} (43, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 260,\ (0:\ ),\ 0.874 + 0.485i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9784674069 + 0.2532061927i$
$L(1)$	$\approx$	$0.9077892554 + 0.1442782781i$

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

Imaginary part of the first few zeros on the critical line

−25.69913456806159381450227411418, −24.70536742159565297632520353199, −23.93656945146979434111970909802, −23.07785947295063932315205234472, −22.487845975055176344686989043921, −21.11577481517626341731309974676, −20.5082542166236833729444038816, −19.18149770729696465126208876305, −18.20118919721379238932032034851, −17.61327800099531850969888973293, −16.67935390136101261654790364586, −15.7389452318837831895162978378, −14.42314369861750074958692200923, −13.589249348355201714126000779290, −12.34130939828071595035973145623, −11.76619457758433848008631097603, −10.57792369036773387109031099303, −9.86537589203947134184724768944, −7.979590946791762763574337687877, −7.518690324482238121394641976671, −6.21316269307819146679988215911, −5.13622902033099253340118794419, −4.23372926085173032092404743981, −2.31030587592929068446473001728, −1.05148758242879048703436585891, 1.147608783590174361403319343733, 2.92847322527736990406610441591, 4.33844012288193807387749355922, 5.376057262575232269654230288, 6.08107366822137129154632221421, 7.584223053601555623999583691032, 8.66062801443701439788287227515, 9.84790313365915777564299920070, 10.884242360779648261713161157516, 11.59570493180116825473564882778, 12.50429477467068802034265277120, 13.82736206807413394617818030151, 14.908118687868812886603667484945, 15.826469531629766994691061758415, 16.60567330403616826958622392147, 17.75044922405493533555558864968, 18.256456726305331541223844120463, 19.474952982180512693871069381383, 20.807401710395696679911142044989, 21.53063864608709471749294988526, 22.06673275423665000814882093837, 23.40875710793788191975355533004, 23.917763135261806986054582338822, 24.88611246082565552647578245458, 26.21725085169859389049401282966

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