L-function 1-1700-1700.1347-r0-0-0

• Label: 1-1700-1700.1347-r0-0-0
• Degree: $1$
• Conductor: $1700$
• Sign: $-0.792 + 0.610i$
• Analytic cond.: $7.89476$
• Root an. cond.: $7.89476$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.309 + 0.951i)3^-s + 7^-s + (−0.809 + 0.587i)9^-s + (−0.587 + 0.809i)11^-s + (0.587 + 0.809i)13^-s + (−0.309 + 0.951i)19^-s + (0.309 + 0.951i)21^-s + (0.809 + 0.587i)23^-s + (−0.809 − 0.587i)27^-s + (−0.951 + 0.309i)29^-s + (0.951 + 0.309i)31^-s + (−0.951 − 0.309i)33^-s + (−0.809 + 0.587i)37^-s + (−0.587 + 0.809i)39^-s + (−0.587 − 0.809i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1700$    =    $2^{2} \cdot 5^{2} \cdot 17$
Sign:	$-0.792 + 0.610i$
Analytic conductor:	$7.89476$
Root analytic conductor:	$7.89476$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1700} (1347, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1700,\ (0:\ ),\ -0.792 + 0.610i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5375000943 + 1.578078073i$
$L(1)$	$\approx$	$1.031110640 + 0.6336477277i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	17	$1$
good	3	$1 + (0.309 + 0.951i)T$
	7	$1 + T$
	11	$1 + (-0.587 + 0.809i)T$
	13	$1 + (0.587 + 0.809i)T$
	19	$1 + (-0.309 + 0.951i)T$
	23	$1 + (0.809 + 0.587i)T$
	29	$1 + (-0.951 + 0.309i)T$
	31	$1 + (0.951 + 0.309i)T$
	37	$1 + (-0.809 + 0.587i)T$

Imaginary part of the first few zeros on the critical line

−20.03689085203902760161211583067, −19.1185185744307978224222385574, −18.63454631382560133496226073074, −17.79664216581436259193979182349, −17.40953263038598092544721914763, −16.41103614572275732013840407532, −15.35148850660935214038271316704, −14.82453877608629715890759136254, −13.92011376498929623592193024610, −13.27679458099166697149126908321, −12.777163380225405810135931850503, −11.63974253653433279686708685857, −11.15295299928311908473917918324, −10.39239109148024714552495651594, −9.02243865369578103576323582676, −8.45789149018707624944389972302, −7.85198912139121290065993794503, −7.08283229806066183952906574214, −6.080613839277258881589509641611, −5.41752696731194053589017739492, −4.4174128330328629726582315230, −3.16223737439585930783072252794, −2.54760923609527422519490130850, −1.45759572657836103282815079996, −0.57106263585010228977407806584, 1.53948568774449537891869136079, 2.26771374571320146965158646772, 3.472130669665002480410041300759, 4.19468136778602908838926461324, 5.00971051543386970728464680314, 5.583474821487107668876265654020, 6.865259514593484155509984969410, 7.76554441547970420066530294319, 8.52723312725154848662248374997, 9.1582939459290657501802208320, 10.1341405089409593880877110332, 10.6900591146431137558851608337, 11.47709857376327333532380030111, 12.20606110522356213614891599143, 13.3609177564599418091646448845, 14.07411284236564671746202005766, 14.72637146450200518955241722646, 15.43093545334150820685399278988, 15.97864213964980207561554516078, 17.12100459051366982831550021820, 17.32162749451126382900349817965, 18.58651263154112337093591843038, 19.01086638378920101737562101256, 20.26522000028625313256926552663, 20.70529554437056846829160382789

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