L-function 1-680-680.99-r0-0-0

• Label: 1-680-680.99-r0-0-0
• Degree: $1$
• Conductor: $680$
• Sign: $0.507 + 0.861i$
• Analytic cond.: $3.15790$
• Root an. cond.: $3.15790$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.923 + 0.382i)3^-s + (0.382 + 0.923i)7^-s + (0.707 + 0.707i)9^-s + (0.923 − 0.382i)11^-s − i·13^-s + (0.707 − 0.707i)19^-s − i·21^-s + (−0.923 + 0.382i)23^-s + (0.382 + 0.923i)27^-s + (0.382 − 0.923i)29^-s + (−0.923 − 0.382i)31^-s + 33^-s + (−0.923 − 0.382i)37^-s + (−0.382 + 0.923i)39^-s + (0.382 + 0.923i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$680$    =    $2^{3} \cdot 5 \cdot 17$
Sign:	$0.507 + 0.861i$
Analytic conductor:	$3.15790$
Root analytic conductor:	$3.15790$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{680} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 680,\ (0:\ ),\ 0.507 + 0.861i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.898812582 + 1.085159231i$
$L(1)$	$\approx$	$1.504225021 + 0.4359191661i$

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	$1 + (0.923 + 0.382i)T$
	7	$1 + (0.382 + 0.923i)T$
	11	$1 + (0.923 - 0.382i)T$
	13	$1 - iT$
	19	$1 + (0.707 - 0.707i)T$
	23	$1 + (-0.923 + 0.382i)T$
	29	$1 + (0.382 - 0.923i)T$
	31	$1 + (-0.923 - 0.382i)T$
	37	$1 + (-0.923 - 0.382i)T$

Imaginary part of the first few zeros on the critical line

−22.67938652403644107910657409010, −21.72158455462695794133550197750, −20.61294783975787240242792858734, −20.11437193103012544667045879395, −19.66477336791960724666403324524, −18.4180198220760929012260273189, −17.85770361831430253778915156564, −16.917409639640546478852934477492, −15.939807187053406182297978827327, −14.90171292296075130289531328956, −14.25438458616154276606360923765, −13.66560941341118973654992445012, −12.61203546317272801485624021934, −11.95330714555306782379533821236, −10.608724985169121560657723736316, −9.94050926501760237414389100534, −8.91404027588460749380482729135, −8.0227437983278026209114162731, −7.323226184107394599188722060114, −6.49699492585795119391823017103, −5.14015768660608953191845669466, −3.92645658131525988397380198689, −3.34513546845489076342838796690, −1.93906179597532448041649873998, −1.06187134536562106210013781757, 1.58245879633643745886045194341, 2.42041813664042527841754523972, 3.55547282536182468811251642163, 4.402371177035344702472447572209, 5.46477970040943145226391651260, 6.59248386981995605198153385992, 7.65707868395384619335871624737, 8.62597537959503215301993553734, 9.20305527661276094856345386867, 9.92259192184355437461065173275, 11.31734929946912320679549608127, 11.81731226075914478568849163340, 13.01462764584449136750832572147, 14.06805016682520641442377757166, 14.40746688757498264634187925692, 15.47580004397535763731025518177, 16.03454446870892622541883554003, 17.05118468860295077528526639666, 18.13237086725818642391038489555, 18.935107468268763537688419778863, 19.60667244037059327838621014110, 20.38418608370430755108320178367, 21.45882938174666402536697810374, 21.7277697989182453352371506018, 22.601757715330085022445887990056

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