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Properties

Label 1-1700-1700.247-r0-0-0
Degree $1$
Conductor $1700$
Sign $-0.468 - 0.883i$
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  + (−0.891 − 0.453i)3-s + (0.707 + 0.707i)7-s + (0.587 + 0.809i)9-s + (0.987 − 0.156i)11-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)19-s + (−0.309 − 0.951i)21-s + (−0.156 − 0.987i)23-s + (−0.156 − 0.987i)27-s + (−0.453 + 0.891i)29-s + (−0.891 + 0.453i)31-s + (−0.951 − 0.309i)33-s + (0.156 − 0.987i)37-s + (0.156 + 0.987i)39-s + (0.156 − 0.987i)41-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)3-s + (0.707 + 0.707i)7-s + (0.587 + 0.809i)9-s + (0.987 − 0.156i)11-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)19-s + (−0.309 − 0.951i)21-s + (−0.156 − 0.987i)23-s + (−0.156 − 0.987i)27-s + (−0.453 + 0.891i)29-s + (−0.891 + 0.453i)31-s + (−0.951 − 0.309i)33-s + (0.156 − 0.987i)37-s + (0.156 + 0.987i)39-s + (0.156 − 0.987i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.468 - 0.883i$
Analytic conductor: \(7.89476\)
Root analytic conductor: \(7.89476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (0:\ ),\ -0.468 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3968200399 - 0.6595588153i\)
\(L(\frac12)\) \(\approx\) \(0.3968200399 - 0.6595588153i\)
\(L(1)\) \(\approx\) \(0.7678387066 - 0.1725274768i\)
\(L(1)\) \(\approx\) \(0.7678387066 - 0.1725274768i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.891 - 0.453i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.987 - 0.156i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (-0.156 - 0.987i)T \)
29 \( 1 + (-0.453 + 0.891i)T \)
31 \( 1 + (-0.891 + 0.453i)T \)
37 \( 1 + (0.156 - 0.987i)T \)
41 \( 1 + (0.156 - 0.987i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (0.987 - 0.156i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 + (0.987 - 0.156i)T \)
79 \( 1 + (0.891 + 0.453i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.891 + 0.453i)T \)
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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

−20.67495419334104370142283875617, −19.88727898763877303687079350544, −19.09841649198350112296755383895, −18.16264842677816084439437504684, −17.43294736484820253025519403807, −16.74101846788599185794581308486, −16.60265625152540702215385697312, −15.17367003885686565280850386898, −14.84354793262745106553284580325, −13.92055593047406842101798529581, −13.04997915269221643549625557418, −12.019524563431649850206944726949, −11.50939303787106764217345312792, −10.93790483217846417484774910866, −9.89013999667879972436173763380, −9.4894847502911043229457259352, −8.35501874136874618919085182715, −7.354331795047572637254090126785, −6.663354742571420219100805664120, −5.882891227403662796571396884700, −4.834597932855937840359084980673, −4.25915171072927533246280227865, −3.59917285161269606560629347200, −1.95487843292813939891523663306, −1.198483946429840694182980539073, 0.32899161909022093407510786835, 1.621195001200077381134897959090, 2.2747323560556824262111400605, 3.60014698987574336711134556385, 4.74991344020139023583860508043, 5.26795267971706401755877944337, 6.18582043032820299267421504106, 6.84768746028110960577287090917, 7.77974515274774793164391944301, 8.56990319764471685502655578365, 9.39537525199234548549026099403, 10.569131983243246729271186137459, 11.0277397399041133393124797626, 11.89318198500224397570663253092, 12.50774371537824094782505720879, 13.03935050176391486192142773811, 14.345853628920783631385079852275, 14.725597409056397503334743709, 15.7080072838632631257372968385, 16.5843807099469837959847280553, 17.19073843175903064881593026316, 17.899457995142382783373414124129, 18.4057953881188956276849200122, 19.308073668050781022384692959644, 19.909715668620878326649805472495

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