L-function 1-1700-1700.771-r0-0-0

• Label: 1-1700-1700.771-r0-0-0
• Degree: $1$
• Conductor: $1700$
• Sign: $-0.0650 + 0.997i$
• 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

+ (−0.649 − 0.760i)3^-s + (0.382 − 0.923i)7^-s + (−0.156 + 0.987i)9^-s + (−0.522 − 0.852i)11^-s + (0.587 + 0.809i)13^-s + (−0.891 + 0.453i)19^-s + (−0.951 + 0.309i)21^-s + (0.522 + 0.852i)23^-s + (0.852 − 0.522i)27^-s + (−0.760 + 0.649i)29^-s + (−0.0784 − 0.996i)31^-s + (−0.309 + 0.951i)33^-s + (−0.522 + 0.852i)37^-s + (0.233 − 0.972i)39^-s + (−0.233 − 0.972i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0650 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1700$    =    $2^{2} \cdot 5^{2} \cdot 17$
Sign:	$-0.0650 + 0.997i$
Analytic conductor:	$7.89476$
Root analytic conductor:	$7.89476$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1700} (771, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1700,\ (0:\ ),\ -0.0650 + 0.997i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2204664175 + 0.2353047618i$
$L(1)$	$\approx$	$0.6976613508 - 0.1635083394i$

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	$1 + (-0.649 - 0.760i)T$
	7	$1 + (0.382 - 0.923i)T$
	11	$1 + (-0.522 - 0.852i)T$
	13	$1 + (0.587 + 0.809i)T$
	19	$1 + (-0.891 + 0.453i)T$
	23	$1 + (0.522 + 0.852i)T$
	29	$1 + (-0.760 + 0.649i)T$
	31	$1 + (-0.0784 - 0.996i)T$
	37	$1 + (-0.522 + 0.852i)T$

Imaginary part of the first few zeros on the critical line

−20.53780181505385163905246344597, −19.368881088970926362138553086862, −18.413965977421122129080922543160, −17.86151553323758300503663465840, −17.29652136318650111021001312626, −16.32853984676868726585856121491, −15.61699823966872538299693831152, −15.05475191112998507634450375517, −14.571640060888981218278125221399, −13.132796593165027666679092528215, −12.59717495499264701041276648648, −11.80426334357907652225972759233, −10.96384431004698281355145845848, −10.441940432400206136869275976391, −9.56104217286850158406456751898, −8.75088297057110741075432349313, −8.07168772520760659644764358063, −6.84154397019229631401788443687, −6.09337012415717075047247211616, −5.15808425838592173418156071798, −4.791258989502444224097760715340, −3.66681155223594240131745086244, −2.71533882277470699270364697689, −1.68831829726445030109168922891, −0.13000971940886768125822671441, 1.20690350028386543739016791465, 1.839549777130026851768439519357, 3.16395287236150262926802569922, 4.1414472176515666622711964827, 5.04908868234665569103269691789, 5.92609634613033735778840540271, 6.62952998690247460368156072249, 7.46311952715205588690100584099, 8.106805793977217490685972304967, 8.959595968853954412377451970513, 10.1738978899748520719038860368, 10.99650170186674301763991862834, 11.28136851747602633102455260198, 12.24286099773699720090324551465, 13.29277001151464522510648502874, 13.513620297291852990650488655486, 14.36024656926075172822870482280, 15.3696058499444760302317299074, 16.51331188044249273232153049716, 16.72262381831180277689123731891, 17.524759913649506396648842193337, 18.40304607959976974496947118753, 18.91523330183098394211943370460, 19.58255918071835429012046103055, 20.59322922416846176391481406433

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