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Properties

Label 1-709-709.14-r1-0-0
Degree $1$
Conductor $709$
Sign $0.872 + 0.488i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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)2-s + (0.710 − 0.703i)3-s + (0.589 + 0.807i)4-s + (0.792 + 0.610i)5-s + (0.952 − 0.305i)6-s + (0.781 − 0.624i)7-s + (0.159 + 0.987i)8-s + (0.00887 − 0.999i)9-s + (0.429 + 0.903i)10-s + (0.560 + 0.828i)11-s + (0.987 + 0.159i)12-s + (−0.364 − 0.931i)13-s + (0.979 − 0.202i)14-s + (0.992 − 0.123i)15-s + (−0.305 + 0.952i)16-s + (−0.0709 + 0.997i)17-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (0.710 − 0.703i)3-s + (0.589 + 0.807i)4-s + (0.792 + 0.610i)5-s + (0.952 − 0.305i)6-s + (0.781 − 0.624i)7-s + (0.159 + 0.987i)8-s + (0.00887 − 0.999i)9-s + (0.429 + 0.903i)10-s + (0.560 + 0.828i)11-s + (0.987 + 0.159i)12-s + (−0.364 − 0.931i)13-s + (0.979 − 0.202i)14-s + (0.992 − 0.123i)15-s + (−0.305 + 0.952i)16-s + (−0.0709 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.633740997 + 1.730037877i\)
\(L(\frac12)\) \(\approx\) \(6.633740997 + 1.730037877i\)
\(L(1)\) \(\approx\) \(2.903982017 + 0.4902197266i\)
\(L(1)\) \(\approx\) \(2.903982017 + 0.4902197266i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (0.710 - 0.703i)T \)
5 \( 1 + (0.792 + 0.610i)T \)
7 \( 1 + (0.781 - 0.624i)T \)
11 \( 1 + (0.560 + 0.828i)T \)
13 \( 1 + (-0.364 - 0.931i)T \)
17 \( 1 + (-0.0709 + 0.997i)T \)
19 \( 1 + (0.355 - 0.934i)T \)
23 \( 1 + (-0.0177 + 0.999i)T \)
29 \( 1 + (0.842 + 0.537i)T \)
31 \( 1 + (0.797 - 0.603i)T \)
37 \( 1 + (-0.624 - 0.781i)T \)
41 \( 1 + (-0.0886 + 0.996i)T \)
43 \( 1 + (0.981 - 0.194i)T \)
47 \( 1 + (-0.484 - 0.874i)T \)
53 \( 1 + (0.921 + 0.388i)T \)
59 \( 1 + (-0.987 + 0.159i)T \)
61 \( 1 + (0.141 + 0.989i)T \)
67 \( 1 + (0.758 - 0.651i)T \)
71 \( 1 + (0.429 - 0.903i)T \)
73 \( 1 + (-0.847 - 0.530i)T \)
79 \( 1 + (-0.445 + 0.895i)T \)
83 \( 1 + (0.678 + 0.734i)T \)
89 \( 1 + (-0.582 - 0.813i)T \)
97 \( 1 + (-0.522 - 0.852i)T \)
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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

−21.98877345659719961019139719779, −21.48514937618009192739766877265, −20.81946318826603053579917745995, −20.39249852352383312104172736493, −19.22133084478605858098846347713, −18.65178202288411899272003536433, −17.27866153817776253777599471985, −16.269429292791440673859904934004, −15.76775566828469383949255631064, −14.38386443676875437484524590608, −14.25206523750837352895362685277, −13.56111788470370433400657123310, −12.27932025085016670991168496808, −11.68210601106800497219987025891, −10.62616711004014943893559229919, −9.72912462970815751514570048281, −9.01150287427351084487766315708, −8.1864403973162852563474632963, −6.664868904440064544047938969568, −5.628330828924485935938132604780, −4.85177070350606416357426814943, −4.20647848966668568510246529064, −2.91181415117015914345966740915, −2.156903226108460821868722801854, −1.164618882224540151382298962713, 1.32091521318395865883398742706, 2.22466852915774553451682266468, 3.14018167000727663969520473420, 4.16134812051415427809382769088, 5.27571057922496437307983362671, 6.30611409332690454826723043794, 7.13640571370472360177868815242, 7.6568290098536914773158333630, 8.68720127398213195730824134095, 9.88315980042854709849147080986, 10.89832472897238948617943924263, 11.93028243962472202635058708539, 12.81407060415073474247898424327, 13.59106501905811313250505457520, 14.115841283627852940948769446355, 14.99864983110259103744100315361, 15.29209955495283352583892374156, 17.01795213754684297555004325366, 17.64904329476094062240429137940, 17.966489038039760047697481996661, 19.62032039516487388525909864879, 20.00243068431963714758273722112, 21.03040198711317025072589870483, 21.60967870306737905716230848971, 22.62824469475402731694247735444

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