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Properties

Label 1-1700-1700.1347-r0-0-0
Degree $1$
Conductor $1700$
Sign $-0.792 + 0.610i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.792 + 0.610i$
Analytic conductor: \(7.89476\)
Root analytic conductor: \(7.89476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (0:\ ),\ -0.792 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5375000943 + 1.578078073i\)
\(L(\frac12)\) \(\approx\) \(0.5375000943 + 1.578078073i\)
\(L(1)\) \(\approx\) \(1.031110640 + 0.6336477277i\)
\(L(1)\) \(\approx\) \(1.031110640 + 0.6336477277i\)

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

−20.03689085203902760161211583067, −19.1185185744307978224222385574, −18.63454631382560133496226073074, −17.79664216581436259193979182349, −17.40953263038598092544721914763, −16.41103614572275732013840407532, −15.35148850660935214038271316704, −14.82453877608629715890759136254, −13.92011376498929623592193024610, −13.27679458099166697149126908321, −12.777163380225405810135931850503, −11.63974253653433279686708685857, −11.15295299928311908473917918324, −10.39239109148024714552495651594, −9.02243865369578103576323582676, −8.45789149018707624944389972302, −7.85198912139121290065993794503, −7.08283229806066183952906574214, −6.080613839277258881589509641611, −5.41752696731194053589017739492, −4.4174128330328629726582315230, −3.16223737439585930783072252794, −2.54760923609527422519490130850, −1.45759572657836103282815079996, −0.57106263585010228977407806584, 1.53948568774449537891869136079, 2.26771374571320146965158646772, 3.472130669665002480410041300759, 4.19468136778602908838926461324, 5.00971051543386970728464680314, 5.583474821487107668876265654020, 6.865259514593484155509984969410, 7.76554441547970420066530294319, 8.52723312725154848662248374997, 9.1582939459290657501802208320, 10.1341405089409593880877110332, 10.6900591146431137558851608337, 11.47709857376327333532380030111, 12.20606110522356213614891599143, 13.3609177564599418091646448845, 14.07411284236564671746202005766, 14.72637146450200518955241722646, 15.43093545334150820685399278988, 15.97864213964980207561554516078, 17.12100459051366982831550021820, 17.32162749451126382900349817965, 18.58651263154112337093591843038, 19.01086638378920101737562101256, 20.26522000028625313256926552663, 20.70529554437056846829160382789

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