L-function 1-260-260.199-r1-0-0

• Label: 1-260-260.199-r1-0-0
• Degree: $1$
• Conductor: $260$
• Sign: $-0.872 + 0.488i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$260$    =    $2^{2} \cdot 5 \cdot 13$
Sign:	$-0.872 + 0.488i$
Analytic conductor:	$27.9408$
Root analytic conductor:	$27.9408$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{260} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 260,\ (1:\ ),\ -0.872 + 0.488i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.1523552296 - 0.5835382032i$
$L(1)$	$\approx$	$0.6798636959 - 0.3809798232i$

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

Imaginary part of the first few zeros on the critical line

−26.04874018034103215118871095450, −25.60450398746883568268317392906, −24.16492385648550905977124712499, −23.44600050534408803396817393610, −22.39209679865342044391604005221, −21.59242420383886086491001460891, −20.95780757593298043284298841890, −19.9339466695050162641137960897, −18.68265175634180261050224401323, −17.66369365692374441842176590604, −17.07162185153835704186333552748, −15.63673328356745084901547048952, −15.31153744125686468248638316723, −14.2815195490394942637969163722, −12.753023066073002914499078382811, −11.92405027641570673500938394905, −10.946714312595929038830047046302, −10.00583930843996880540097511233, −9.039239134686433173707545308537, −7.976221643211425112707194261075, −6.48760796247073842570276753882, −5.36863894835302847853239087018, −4.641225142731184130706699795099, −3.28770032081180768434747541368, −1.83444993313955710189602597387, 0.20593116045547772811276918329, 1.34312685120691394804881584135, 2.771848633028950678100265449203, 4.34633039606668098765945203172, 5.5588359260136902607218494205, 6.537693054544142320904438988848, 7.71628307264455985298500489192, 8.28888558035901104377953133556, 10.02850615530129226377404318078, 10.97561192112617959275976588016, 11.78437218431059236589076065037, 12.90102835129457714593993998332, 13.781144096575271275896096957052, 14.49634358617532637891611865679, 16.17126287554210636308150391101, 16.77919726225042222222529922977, 17.79632298776074252718782515488, 18.60311747040356251377175435852, 19.425452577794739641796219015805, 20.57083709984608607812608702767, 21.39093010316908048757721024011, 22.82793295638804362714338361845, 23.19858592744874922999903937723, 24.335737124544114219894248810899, 24.75557346196074694114000303510

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