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Properties

Label 1-709-709.22-r1-0-0
Degree $1$
Conductor $709$
Sign $0.998 + 0.0595i$
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  + (0.946 − 0.322i)2-s + (0.997 + 0.0709i)3-s + (0.792 − 0.610i)4-s + (0.468 + 0.883i)5-s + (0.967 − 0.254i)6-s + (−0.202 + 0.979i)7-s + (0.552 − 0.833i)8-s + (0.989 + 0.141i)9-s + (0.728 + 0.684i)10-s + (0.996 − 0.0886i)11-s + (0.833 − 0.552i)12-s + (0.314 − 0.949i)13-s + (0.123 + 0.992i)14-s + (0.405 + 0.914i)15-s + (0.254 − 0.967i)16-s + (0.906 − 0.421i)17-s + ⋯
L(s)  = 1  + (0.946 − 0.322i)2-s + (0.997 + 0.0709i)3-s + (0.792 − 0.610i)4-s + (0.468 + 0.883i)5-s + (0.967 − 0.254i)6-s + (−0.202 + 0.979i)7-s + (0.552 − 0.833i)8-s + (0.989 + 0.141i)9-s + (0.728 + 0.684i)10-s + (0.996 − 0.0886i)11-s + (0.833 − 0.552i)12-s + (0.314 − 0.949i)13-s + (0.123 + 0.992i)14-s + (0.405 + 0.914i)15-s + (0.254 − 0.967i)16-s + (0.906 − 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.998 + 0.0595i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.998 + 0.0595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.511697152 + 0.2237657981i\)
\(L(\frac12)\) \(\approx\) \(7.511697152 + 0.2237657981i\)
\(L(1)\) \(\approx\) \(3.184361910 - 0.05915793624i\)
\(L(1)\) \(\approx\) \(3.184361910 - 0.05915793624i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.946 - 0.322i)T \)
3 \( 1 + (0.997 + 0.0709i)T \)
5 \( 1 + (0.468 + 0.883i)T \)
7 \( 1 + (-0.202 + 0.979i)T \)
11 \( 1 + (0.996 - 0.0886i)T \)
13 \( 1 + (0.314 - 0.949i)T \)
17 \( 1 + (0.906 - 0.421i)T \)
19 \( 1 + (0.895 - 0.445i)T \)
23 \( 1 + (-0.280 + 0.959i)T \)
29 \( 1 + (-0.943 + 0.330i)T \)
31 \( 1 + (-0.807 - 0.589i)T \)
37 \( 1 + (-0.979 - 0.202i)T \)
41 \( 1 + (-0.988 + 0.150i)T \)
43 \( 1 + (0.999 + 0.0177i)T \)
47 \( 1 + (0.237 + 0.971i)T \)
53 \( 1 + (0.106 - 0.994i)T \)
59 \( 1 + (-0.833 - 0.552i)T \)
61 \( 1 + (-0.764 + 0.645i)T \)
67 \( 1 + (0.355 + 0.934i)T \)
71 \( 1 + (0.728 - 0.684i)T \)
73 \( 1 + (0.461 + 0.887i)T \)
79 \( 1 + (0.891 - 0.453i)T \)
83 \( 1 + (0.596 - 0.802i)T \)
89 \( 1 + (0.492 + 0.870i)T \)
97 \( 1 + (-0.582 + 0.813i)T \)
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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

−22.358094065000952466064504953426, −21.465620990901306396046714591655, −20.741891785214561085742833477400, −20.2365064484552232220322790019, −19.56028634496390743194976350880, −18.42142594794057015160982584824, −16.9066361605492801946000337798, −16.74192003385845603432622507895, −15.80465617496943768638075663000, −14.66262332937886382640785930998, −13.90484503802624075542023985327, −13.728345932986660260824941124841, −12.54706815149174076441707250885, −12.077417861188194868503963557610, −10.65669290123807991538203235728, −9.625202674433820632931759823128, −8.803553511019095128971579449562, −7.82745621992430432185442327742, −6.99780098642928478953662206845, −6.131007804394692981856258704226, −4.90383003641675066597051428468, −3.94327675639065055035929943775, −3.51778424613212360839922128258, −1.94067188437579564481057700592, −1.25907192550061027168314061755, 1.36854402335828639237199345140, 2.292218196050998854423559850319, 3.29806917986169823218062507242, 3.56096199785584619023298755392, 5.23041044456277633649648019070, 5.90272695587708472869714339020, 6.98248995746708493397594456798, 7.76857622861230633762921111663, 9.28331120293995022561127665699, 9.68218342648233357188494234402, 10.787607387471239896554745197532, 11.70455717965134439357347850344, 12.59527275623507555367039426455, 13.51604583475816290328794782256, 14.14104207601986420069469921433, 14.8851725699626168673819331984, 15.413934059456776175788808287, 16.27199491102881053392349690291, 17.78511783802470417223731550573, 18.70259251336427572679513173339, 19.22501667432322869087101952246, 20.15247307550909888057881919801, 20.84852585921636743956784047083, 21.785019453383055619654051085124, 22.199182448821686453177382752716

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