L-function 1-2557-2557.4-r0-0-0

• Label: 1-2557-2557.4-r0-0-0
• Degree: $1$
• Conductor: $2557$
• Sign: $-0.690 - 0.723i$
• 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

+ (0.999 + 0.00491i)2^-s + (0.648 − 0.761i)3^-s + (0.999 + 0.00983i)4^-s + (0.614 − 0.789i)5^-s + (0.652 − 0.758i)6^-s + (−0.817 − 0.576i)7^-s + (0.999 + 0.0147i)8^-s + (−0.159 − 0.987i)9^-s + (0.617 − 0.786i)10^-s + (0.134 − 0.990i)11^-s + (0.655 − 0.754i)12^-s + (−0.985 − 0.171i)13^-s + (−0.814 − 0.580i)14^-s + (−0.202 − 0.979i)15^-s + (0.999 + 0.0196i)16^-s + (0.988 + 0.151i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2557$
Sign:	$-0.690 - 0.723i$
Analytic conductor:	$11.8746$
Root analytic conductor:	$11.8746$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2557} (4, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2557,\ (0:\ ),\ -0.690 - 0.723i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.668264810 - 3.896881111i$
$L(1)$	$\approx$	$2.004164902 - 1.392826673i$

Euler product

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

	$p$	$F_p(T)$
bad	2557	$1$
good	2	$1 + (0.999 + 0.00491i)T$
	3	$1 + (0.648 - 0.761i)T$
	5	$1 + (0.614 - 0.789i)T$
	7	$1 + (-0.817 - 0.576i)T$
	11	$1 + (0.134 - 0.990i)T$
	13	$1 + (-0.985 - 0.171i)T$
	17	$1 + (0.988 + 0.151i)T$
	19	$1 + (-0.929 - 0.369i)T$
	23	$1 + (0.967 - 0.252i)T$

Imaginary part of the first few zeros on the critical line

−19.67239252023361582349264703524, −19.2488243596833274607405707377, −18.51268786382191993332060186314, −17.12960400129567420976092041449, −16.8338778370091069897361502976, −15.70678061241716574551243863061, −15.2591865677899005713211552410, −14.60533010407539572210898936986, −14.21763487534989042231401365092, −13.35638500902341342612022373195, −12.5162043096290098090112337118, −12.064933131414859760024022125188, −10.77871649379823059733516668489, −10.38132904216408443804879128247, −9.58611126037841920107630414710, −9.0684676326784875717749720340, −7.594386823175012904747993206149, −7.19629885717870946301821732879, −6.19584350296141030901238224715, −5.50587041045922351759549979420, −4.731724125304093949687138158203, −3.83839137592894868791043028570, −3.088507111505972388808206904332, −2.42340012278757643420062416882, −1.89154013408296667953865415499, 0.79149579181814517395364459192, 1.52690364603584035618325531047, 2.73205036402955651432583527744, 3.07477180833036111540537423887, 4.11782831089924712481882237106, 4.97162151152862919838547726137, 5.931615409208131142406732385452, 6.45025061279793367793526569062, 7.23144218866154252699715865678, 8.03359812836392116925257793793, 8.85763343825434379481793204834, 9.722012509311931843926164924069, 10.45507178762804994044777863831, 11.49332702234859299158514458976, 12.38698049849266091492096450599, 12.96705921401333495949049764852, 13.16328990116521928095937608719, 14.17840667017442629051034531067, 14.458093593206895717672421279511, 15.41131579097921835493652685213, 16.46284164734002789066604717642, 16.78878973852052001785924863466, 17.50944672731212095102155497562, 18.769126548860940978121882486711, 19.36445207264607870798680219530

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