L-function 1-2557-2557.1061-r0-0-0

• Label: 1-2557-2557.1061-r0-0-0
• Degree: $1$
• Conductor: $2557$
• Sign: $-0.246 + 0.969i$
• Analytic cond.: $11.8746$
• Root an. cond.: $11.8746$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.805 − 0.592i)2^-s + (0.0957 − 0.995i)3^-s + (0.297 − 0.954i)4^-s + (−0.461 − 0.887i)5^-s + (−0.512 − 0.858i)6^-s + (0.740 − 0.672i)7^-s + (−0.325 − 0.945i)8^-s + (−0.981 − 0.190i)9^-s + (−0.897 − 0.441i)10^-s + (−0.986 + 0.161i)11^-s + (−0.921 − 0.387i)12^-s + (0.978 − 0.205i)13^-s + (0.197 − 0.980i)14^-s + (−0.927 + 0.374i)15^-s + (−0.822 − 0.568i)16^-s + (0.688 − 0.725i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2557$
Sign:	$-0.246 + 0.969i$
Analytic conductor:	$11.8746$
Root analytic conductor:	$11.8746$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2557} (1061, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2557,\ (0:\ ),\ -0.246 + 0.969i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-1.271906506 - 1.635667288i$
$L(1)$	$\approx$	$0.6305458972 - 1.354346682i$

Euler product

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

	$p$	$F_p(T)$
bad	2557	$1$
good	2	$1 + (0.805 - 0.592i)T$
	3	$1 + (0.0957 - 0.995i)T$
	5	$1 + (-0.461 - 0.887i)T$
	7	$1 + (0.740 - 0.672i)T$
	11	$1 + (-0.986 + 0.161i)T$
	13	$1 + (0.978 - 0.205i)T$
	17	$1 + (0.688 - 0.725i)T$
	19	$1 + (-0.139 - 0.990i)T$
	23	$1 + (0.00737 + 0.999i)T$

Imaginary part of the first few zeros on the critical line

−20.456111254298998609970501226685, −18.90676707110430182784775528876, −18.61678090666129486087255493261, −17.62768897292422720515387199198, −16.80413877058883185512653826396, −16.09379759351874230727529143109, −15.47688973935338545973597075138, −14.995704297342950821737740342696, −14.41623823329533446564302253441, −13.84106290575086105708868989467, −12.7921518146208174554110080833, −11.9442635789026162468206237251, −11.24051233955535958421794712160, −10.73973107048317310014743967293, −9.92729117858364988444153637393, −8.59922197983375430222498547134, −8.1971830959903912379054979122, −7.628193914173382345401263138752, −6.23576722916451579694917592142, −5.92282972543232145572176351231, −5.00345716007701344862679042149, −4.24351201249373494978925534664, −3.500358806779807816996972774779, −2.86718589344212225844152218787, −1.97146017751647439430393176030, 0.48379469348741593380875525711, 1.23008455050936982561905629287, 1.95723005399891006617786637739, 3.04862651592018851652396489620, 3.76153174309576173232471945882, 4.868697218556752824870460082046, 5.257732990243409277080446966715, 6.15752157328028667653234701456, 7.293087135488528085190933255993, 7.69830346771039845287549020593, 8.587916999312304765759918307178, 9.44099684992124895110307037130, 10.52138159851199452953829069160, 11.393698647490577795910982710399, 11.59589305952323007173050254174, 12.657341611815884992364199674, 13.14198085659723168711931247966, 13.635212869369182295234646236804, 14.30513660142289538825558575798, 15.29048097041701582604813942861, 15.876451821385419286688083661267, 16.7706055734043288395221381736, 17.74541263212694739437983710339, 18.23752301131083865059069378031, 19.12043836432022246551198087215

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