L-function 1-680-680.587-r0-0-0

• Label: 1-680-680.587-r0-0-0
• Degree: $1$
• Conductor: $680$
• Sign: $0.564 - 0.825i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$680$    =    $2^{3} \cdot 5 \cdot 17$
Sign:	$0.564 - 0.825i$
Analytic conductor:	$3.15790$
Root analytic conductor:	$3.15790$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{680} (587, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 680,\ (0:\ ),\ 0.564 - 0.825i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.809411638 - 0.9543282547i$
$L(1)$	$\approx$	$1.424106181 - 0.4025996502i$

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

Imaginary part of the first few zeros on the critical line

−22.72267180065815144385605668069, −21.89521143521862238867371270904, −21.181284281230577727014599377395, −20.34019312633147841777660481540, −19.87221716644958674144198062379, −18.94074130257173124653938186825, −17.884209953442823814465614017246, −16.89756761645819009356124065573, −16.4190442743259816403040486991, −15.16507434908636343445346805760, −14.65439395790294386209675231504, −13.92230826843447010744403618576, −13.09792905606959415869251427086, −11.82095810548263246157669174178, −10.9808256130885753210044353222, −10.22098076644824289992108410086, −9.1435025633020552991361267542, −8.70036153947009079873912225376, −7.38509535620992576085859937595, −6.85658595983827334934119026290, −5.17319371007825785986332662974, −4.4344096500816258055723220331, −3.74250058574302464252050715711, −2.43272917815265278013874037526, −1.42958386536690474258121719637, 1.04477343671452807210959399385, 2.04830605385617402770962754432, 3.089923970784832133055245452822, 4.00560921531938509613222364851, 5.53232236039583799491496012885, 6.121470278918336553693172826429, 7.46823664871464463011188549741, 8.06010147640169449150067900527, 8.8995153738192737053896535730, 9.67168943399954287892946946403, 11.06851509924264129837285543636, 11.77303976028575209621546265494, 12.705325310271392588531680152863, 13.412116482001346133268493028740, 14.511584988878100210559829636828, 14.83901423337132772118371136082, 15.85657955733458957286366321133, 17.07896396266266489049399602607, 17.80218217208029380404293603112, 18.681911442818088975443860814001, 19.17535178530898115419850717065, 20.197396724593316834449140631499, 20.84079380561629345065828839495, 21.69360794266771169951139397852, 22.62792861406981818326236245031

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