L-function 1-2856-2856.251-r1-0-0

• Label: 1-2856-2856.251-r1-0-0
• Degree: $1$
• Conductor: $2856$
• Sign: $-0.615 - 0.788i$
• Analytic cond.: $306.919$
• Root an. cond.: $306.919$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− i·5^-s + i·11^-s + 13^-s + 19^-s − i·23^-s − 25^-s + i·29^-s + i·31^-s − i·37^-s − i·41^-s − 43^-s − 47^-s − 53^-s + 55^-s − 59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2856$    =    $2^{3} \cdot 3 \cdot 7 \cdot 17$
Sign:	$-0.615 - 0.788i$
Analytic conductor:	$306.919$
Root analytic conductor:	$306.919$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2856} (251, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2856,\ (1:\ ),\ -0.615 - 0.788i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.6382674128 - 1.308117048i$
$L(1)$	$\approx$	$1.030780644 - 0.2081568906i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	7	$1$
	17	$1$
good	5	$1$
	11	$1$
	13	$1 - iT$
	19	$1$
	23	$1$
	29	$1$
	31	$1$
	37	$1 + iT$
	41	$1$

Imaginary part of the first few zeros on the critical line

−19.04350259155210387469606582461, −18.59754326504966725642136250292, −18.00139606927771160997748690447, −17.19735924971810713794316997633, −16.35636082794606903095842735139, −15.676021105178687464445767208135, −15.09543559747064871493650444482, −14.24114826862400162687206758004, −13.5043509013046001567864472023, −13.27077099288747064555497560828, −11.7262072013693717845264719434, −11.502596610431207548647541541941, −10.799523305920070372607583816365, −9.90591122146893173231658396061, −9.346192639273855224358538941244, −8.18571655819844541866446286927, −7.81443538928999259052059214885, −6.75450796250562573608609391725, −6.12497816266200419835880663788, −5.53441550210924362969207628829, −4.369068202254887426661908872241, −3.31534165113142018019201093758, −3.13329463361873496925804551453, −1.870861246599765833432899955303, −0.94068028548101282018276012688, 0.255483636365749693962321389457, 1.294224941774004523402976266549, 1.89147995854238812105401664907, 3.14401932069365511950510396778, 3.95425614451248486028912311316, 4.84991334650559685392153142653, 5.3156013156881497896339884218, 6.34494615488167318032749200303, 7.112379417989166453422015116890, 7.99189326219176594033082499266, 8.69409264809243750234479583631, 9.30976336919169382867180896447, 10.077391148069206690965036957758, 10.89752459946952280826505617706, 11.72448311765797687036511206721, 12.60006295789862133055913872285, 12.78383541086347326526944072825, 13.86855282689577806755137328881, 14.36916577247828930816345579317, 15.45013451831295054300190273699, 15.970055665915522402358835068175, 16.54425239207520863322644964187, 17.37608214120733314983511067500, 18.03755528976506984484603493164, 18.5858930121901638750133404024

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