L-function 1-5e2-25.13-r1-0-0

• Label: 1-5e2-25.13-r1-0-0
• Degree: $1$
• Conductor: $25$
• Sign: $0.187 - 0.982i$
• Analytic cond.: $2.68662$
• Root an. cond.: $2.68662$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.951 − 0.309i)2^-s + (−0.587 − 0.809i)3^-s + (0.809 − 0.587i)4^-s + (−0.809 − 0.587i)6^-s − i·7^-s + (0.587 − 0.809i)8^-s + (−0.309 + 0.951i)9^-s + (0.309 + 0.951i)11^-s + (−0.951 − 0.309i)12^-s + (0.951 + 0.309i)13^-s + (−0.309 − 0.951i)14^-s + (0.309 − 0.951i)16^-s + (−0.587 + 0.809i)17^-s + i·18^-s + (0.809 + 0.587i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$25$    =    $5^{2}$
Sign:	$0.187 - 0.982i$
Analytic conductor:	$2.68662$
Root analytic conductor:	$2.68662$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{25} (13, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 25,\ (1:\ ),\ 0.187 - 0.982i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.482218465 - 1.226197754i$
$L(1)$	$\approx$	$1.378802374 - 0.7129109418i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
good	2	$1 + (0.951 - 0.309i)T$
	3	$1 + (-0.587 - 0.809i)T$
	7	$1 - iT$
	11	$1 + (0.309 + 0.951i)T$
	13	$1 + (0.951 + 0.309i)T$
	17	$1 + (-0.587 + 0.809i)T$
	19	$1 + (0.809 + 0.587i)T$
	23	$1 + (-0.951 + 0.309i)T$
	29	$1 + (0.809 - 0.587i)T$

Imaginary part of the first few zeros on the critical line

−38.4452568534993322207135798615, −37.74583052648439184978635236180, −35.148544981062632168677880280817, −34.430937675677178326012078771082, −33.12199662141554282205170977353, −32.19829178912301325466792523076, −31.01676168048359971116478493543, −29.3956505188558257526594211547, −28.20148379648412894448495556373, −26.63347539267659867738715781851, −25.14731898512247250354570849926, −23.80309499718615968431776583302, −22.35576466016837459450057651849, −21.681153704098747260886540395419, −20.326983525217915866082945291589, −18.01987117849817655242877267313, −16.243900265057743914117467792210, −15.5407910552972031602988945058, −13.93504548429567351049649728926, −12.09271183978261830717139328509, −11.01035637784640690626539889546, −8.78700865446590678715430717171, −6.3107254305470667701549426079, −5.12731131686940929145413272001, −3.299412304185740963460154106670, 1.58220676000882298958522643392, 4.159560687065339514036455869063, 6.0762064126980081679370655263, 7.39593930121967567884613681378, 10.4178092774473160463202280021, 11.74692761578520974197903825592, 13.07143163162304960966022768946, 14.167424483808897243122084319674, 16.11990570142744652770830929631, 17.67324902540025821502366222951, 19.404237458897755627662812147778, 20.58085192224977800941300201367, 22.37283659680972527442878489317, 23.31070693443765994536139747628, 24.270126704651321674008307671680, 25.74427898855974964160548641100, 28.00979912702797634246984899424, 29.10355236216309161927950165122, 30.27505466527500294819961545390, 30.994475497945596764249261700722, 32.94093397731409517051495856432, 33.63428978147748324985848301743, 35.25730937377627798835378689423, 36.470756307170771026140700066559, 38.09866824913236886118927208397

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