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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.2204664175 + 0.2353047618i\)
\(L(\frac12)\) \(\approx\) \(0.2204664175 + 0.2353047618i\)
\(L(1)\) \(\approx\) \(0.6976613508 - 0.1635083394i\)
\(L(1)\) \(\approx\) \(0.6976613508 - 0.1635083394i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.649 - 0.760i)T \)
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 \( 1 + (-0.522 + 0.852i)T \)
41 \( 1 + (-0.233 - 0.972i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.453 + 0.891i)T \)
59 \( 1 + (-0.156 + 0.987i)T \)
61 \( 1 + (-0.852 + 0.522i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.649 - 0.760i)T \)
73 \( 1 + (-0.233 + 0.972i)T \)
79 \( 1 + (-0.0784 + 0.996i)T \)
83 \( 1 + (0.891 - 0.453i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (-0.760 + 0.649i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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