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Properties

Label 1-709-709.138-r0-0-0
Degree $1$
Conductor $709$
Sign $0.529 + 0.848i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (0.977 − 0.211i)2-s + (0.484 + 0.874i)3-s + (0.910 − 0.413i)4-s + (0.861 + 0.507i)5-s + (0.658 + 0.752i)6-s + (0.0797 + 0.996i)7-s + (0.802 − 0.596i)8-s + (−0.530 + 0.847i)9-s + (0.949 + 0.314i)10-s + (−0.237 + 0.971i)11-s + (0.802 + 0.596i)12-s + (0.0797 − 0.996i)13-s + (0.288 + 0.957i)14-s + (−0.0266 + 0.999i)15-s + (0.658 − 0.752i)16-s + (−0.237 − 0.971i)17-s + ⋯
L(s)  = 1  + (0.977 − 0.211i)2-s + (0.484 + 0.874i)3-s + (0.910 − 0.413i)4-s + (0.861 + 0.507i)5-s + (0.658 + 0.752i)6-s + (0.0797 + 0.996i)7-s + (0.802 − 0.596i)8-s + (−0.530 + 0.847i)9-s + (0.949 + 0.314i)10-s + (−0.237 + 0.971i)11-s + (0.802 + 0.596i)12-s + (0.0797 − 0.996i)13-s + (0.288 + 0.957i)14-s + (−0.0266 + 0.999i)15-s + (0.658 − 0.752i)16-s + (−0.237 − 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.529 + 0.848i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.529 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.188077369 + 1.768508781i\)
\(L(\frac12)\) \(\approx\) \(3.188077369 + 1.768508781i\)
\(L(1)\) \(\approx\) \(2.344300443 + 0.7206309531i\)
\(L(1)\) \(\approx\) \(2.344300443 + 0.7206309531i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.977 - 0.211i)T \)
3 \( 1 + (0.484 + 0.874i)T \)
5 \( 1 + (0.861 + 0.507i)T \)
7 \( 1 + (0.0797 + 0.996i)T \)
11 \( 1 + (-0.237 + 0.971i)T \)
13 \( 1 + (0.0797 - 0.996i)T \)
17 \( 1 + (-0.237 - 0.971i)T \)
19 \( 1 + (0.802 - 0.596i)T \)
23 \( 1 + (-0.437 - 0.899i)T \)
29 \( 1 + (-0.339 - 0.940i)T \)
31 \( 1 + (-0.0266 + 0.999i)T \)
37 \( 1 + (0.0797 + 0.996i)T \)
41 \( 1 + (-0.769 + 0.638i)T \)
43 \( 1 + (-0.964 - 0.263i)T \)
47 \( 1 + (-0.437 - 0.899i)T \)
53 \( 1 + (-0.0266 + 0.999i)T \)
59 \( 1 + (0.802 - 0.596i)T \)
61 \( 1 + (-0.887 - 0.461i)T \)
67 \( 1 + (0.734 - 0.678i)T \)
71 \( 1 + (0.949 - 0.314i)T \)
73 \( 1 + (-0.617 + 0.786i)T \)
79 \( 1 + (-0.697 - 0.716i)T \)
83 \( 1 + (-0.987 - 0.159i)T \)
89 \( 1 + (-0.132 + 0.991i)T \)
97 \( 1 + (0.994 - 0.106i)T \)
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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

−22.54560646251254656606214156462, −21.524949564903066207792914583699, −20.98723996369833023925191052304, −20.14752421351326028854252779072, −19.51184898422029304788670615607, −18.417303537373176945112362079674, −17.37579125965597485696589749135, −16.738969277065445111277825429692, −15.992067624990877303620300495453, −14.54272656126597693653513589583, −14.10509811548887541841568914233, −13.36561998070252443129253882760, −12.987602235794916449245519637900, −11.87024578562473381852040516075, −11.05903582973757118008163263922, −9.88939898235660473036987867983, −8.710443499360628833364194183990, −7.86644751363119544583482161853, −6.97458346061200004796623544118, −6.12758007862125597567600016461, −5.42320611932302203839943801130, −4.06431691263311012645036274250, −3.32197948528538770079932665209, −1.979195487047399273636183217321, −1.33009473585352556777785786491, 1.9186375917568258719114146256, 2.6952921470950823078136074549, 3.261427889558464210844685141718, 4.82817036197383101587264516408, 5.155008431987186707769970708773, 6.17770830244899404900139335344, 7.24022658052172485833026680605, 8.44360726190857837883679944782, 9.72452107819451438814618596383, 10.027337218986427050025177815197, 11.07957777054415398876546908834, 11.93638986400932300742283942259, 13.00496604515587501707763292817, 13.74849560496251954610064655964, 14.53923381792510884789323266520, 15.3470555638570761290322503134, 15.612930765864698377563209016269, 16.81905795522268635405138114642, 17.99245787307493332580348186218, 18.669100290626510040677120262172, 20.06971146588493700904774408448, 20.3865119221505097831044570137, 21.27147892879676193297922006409, 21.9895534782022160593394631818, 22.45288774961466597490238729216

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