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Properties

Label 1-1700-1700.1203-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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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: $\chi_{1700} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (0:\ ),\ 0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647549418 + 1.078498284i\)
\(L(\frac12)\) \(\approx\) \(1.647549418 + 1.078498284i\)
\(L(1)\) \(\approx\) \(1.243243384 + 0.4099743356i\)
\(L(1)\) \(\approx\) \(1.243243384 + 0.4099743356i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.309 + 0.951i)T \)
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 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (0.951 + 0.309i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−19.94466812435605777028480222093, −19.5868833365150786552175354400, −18.6708528578099921150072776989, −17.97658523324047172961357559112, −17.33721342988823653837327151316, −16.83949459280930215426654960256, −15.54222074601699309932973157529, −14.684181212702724559392323021035, −14.33481525715943481958688559882, −13.54956600951863788881799234444, −12.54269866532079818839560154295, −12.08538776443324139129711578053, −11.30133809894985422087491850560, −10.49792916519299211651414266990, −9.15277883405182605631142679138, −8.86281902110498020889633149262, −7.83985361074543305297670225777, −6.97127842014123950707240322557, −6.72888090148258571561812277758, −5.33010748450648977183901324491, −4.652988045888649946955767626620, −3.63284125406587680551302936032, −2.31026608791170889717871672381, −1.92724549300279486889347668172, −0.79833751531208440989952239601, 1.05860797462684448818286376055, 2.24577721962499503609173108187, 3.2337134236932346310547551459, 3.94792479096052211280201635396, 4.93901164218368117722450778456, 5.45143812806124083745835371397, 6.46824377456453349199943403501, 7.807027630248599585545681512952, 8.20947585881028771202807522522, 9.066475214100568068128313414735, 9.86498366840490895717545004175, 10.66718252192832884529671844983, 11.28749388343281979751101793302, 11.999963647844752476960411580504, 13.109634035559455722938683793314, 14.00143533462487573804839632514, 14.66373284963568009004269328199, 15.05714048227349206201103773553, 16.02966743421545907500231910641, 16.76739171892933429915012892185, 17.3510438907005544629180051171, 18.15024180699077982215687246130, 19.22824954104745092050403120658, 19.75945279957506908540641350586, 20.595553240657421062491821141819

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