L-function 1-1275-1275.56-r0-0-0

• Label: 1-1275-1275.56-r0-0-0
• Degree: $1$
• Conductor: $1275$
• Sign: $0.914 - 0.403i$
• Analytic cond.: $5.92107$
• Root an. cond.: $5.92107$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.156 + 0.987i)2^-s + (−0.951 − 0.309i)4^-s + (−0.923 + 0.382i)7^-s + (0.453 − 0.891i)8^-s + (0.852 + 0.522i)11^-s + (0.587 − 0.809i)13^-s + (−0.233 − 0.972i)14^-s + (0.809 + 0.587i)16^-s + (−0.891 − 0.453i)19^-s + (−0.649 + 0.760i)22^-s + (−0.852 − 0.522i)23^-s + (0.707 + 0.707i)26^-s + (0.996 − 0.0784i)28^-s + (−0.649 + 0.760i)29^-s + (0.996 + 0.0784i)31^-s + (−0.707 + 0.707i)32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1275$    =    $3 \cdot 5^{2} \cdot 17$
Sign:	$0.914 - 0.403i$
Analytic conductor:	$5.92107$
Root analytic conductor:	$5.92107$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1275} (56, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1275,\ (0:\ ),\ 0.914 - 0.403i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7242338796 - 0.1528004856i$
$L(1)$	$\approx$	$0.7214116493 + 0.2519070505i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
	17	$1$
good	2	$1 + (-0.156 + 0.987i)T$
	7	$1 + (-0.923 + 0.382i)T$
	11	$1 + (0.852 + 0.522i)T$
	13	$1 + (0.587 - 0.809i)T$
	19	$1 + (-0.891 - 0.453i)T$
	23	$1 + (-0.852 - 0.522i)T$
	29	$1 + (-0.649 + 0.760i)T$
	31	$1 + (0.996 + 0.0784i)T$
	37	$1 + (0.852 - 0.522i)T$

Imaginary part of the first few zeros on the critical line

−21.066282081426468270728867398472, −20.22343154501601689536129677347, −19.51131737500405113203365595957, −18.989462585228930163191299997228, −18.344994698126191159880579432138, −17.1474404580248739327097333988, −16.78873840037796637847179564866, −15.89450628422444827804595157670, −14.70996077292483253643042465059, −13.82038813061225890128330011751, −13.3681414487279165998401532128, −12.48023766769978974083055878042, −11.62760156681784980027892358097, −11.1027287380198123038930620303, −9.97473972658307018047759767509, −9.58894502666391493863663007383, −8.63709646020267018895701261967, −7.90199207222382125011580408620, −6.550085731302085891099282463514, −6.02325125897151710853521316818, −4.55016546898034438067687681645, −3.86023453624702865653672590402, −3.20268019756680508411078484664, −2.00889891944505565587447782277, −1.07851418654060361260461527182, 0.356782833845925009300102919711, 1.79151099764796963583260529205, 3.20928168386562959209694122616, 4.057601393854705656783458621522, 5.01130613731924353498154273623, 6.08742765506418446838770673355, 6.48101088712402419573074225435, 7.377921662716530513875993772737, 8.44461882855025023431280037939, 8.96915882425127331003694280737, 9.89286990742307280180680403437, 10.493997997023017485263168527076, 11.81860590527264579155351620746, 12.73791176040775783726112325826, 13.25512291938481073511836153429, 14.21210007355640620370571151253, 15.09671330447184541229798734536, 15.50397756336146046613723996053, 16.47770020593289936492121252617, 16.94939275600107454643912449311, 17.985760330821715363340319490246, 18.40678742809026264638849302519, 19.503212623460937111088645479408, 19.859070490057748239691303430060, 21.08482529671299879471551254182

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