L-function 1-1700-1700.1483-r0-0-0

• Label: 1-1700-1700.1483-r0-0-0
• Degree: $1$
• Conductor: $1700$
• Sign: $-0.400 + 0.916i$
• 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

$\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1700$    =    $2^{2} \cdot 5^{2} \cdot 17$
Sign:	$-0.400 + 0.916i$
Analytic conductor:	$7.89476$
Root analytic conductor:	$7.89476$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1700} (1483, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1700,\ (0:\ ),\ -0.400 + 0.916i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5277714601 + 0.8062410247i$
$L(1)$	$\approx$	$0.7816424790 + 0.3056687227i$

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

−19.825138873233011888661592265478, −19.26449274090803824311803037523, −18.66156035986173348163406698406, −18.09971448012830159899970509867, −16.98847369552191340571391997119, −16.48173868139703741151999380134, −15.977199741877566118273820397341, −14.64406670889991897722750998755, −13.95632124744096002393064521853, −13.39734919377175376926089416810, −12.506484672367041335101215556819, −11.99716495157664542864593094972, −11.14832014280797215828138097081, −10.38127691853272295861040597238, −9.30565795635541385800380779133, −8.61932414050737292984009093445, −7.781701939205178834394399802643, −6.82740959126941995533035514253, −6.12453210221439996787188532921, −5.828925687420782199652117164267, −4.308446945941141431690924203261, −3.511493681370436592723906483713, −2.484474882433385713104637279127, −1.53856484584931261190631074337, −0.44614875644587987638647116073, 0.96154207787641037583288489437, 2.49091775990808348548938798752, 3.360362892114364883135971891926, 4.078383195182941889504536740241, 4.919201169909633747666481742021, 5.90192639640033909386478782743, 6.47687648252087121404238605033, 7.45543292363254265460551814033, 8.618403136479526012811492808985, 9.3223401209046672370981053360, 9.93126452397073537149837171489, 10.64507157884427254836116326717, 11.42192734716910085752720043636, 12.33133519652050514292080522710, 12.9532342450965342347426653957, 13.93923553812098231560793892046, 14.78330766323612713081048300433, 15.59320652655825302378965047954, 15.934281585692468558947584868380, 16.79736876573565677087352339889, 17.58587187693948467845879169501, 18.0833006902148632292812846424, 19.325095611929741817892752384506, 19.990964981908392542283104952417, 20.393526491065592133626997807

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