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Properties

Label 1-709-709.186-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.503 - 0.864i$
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  + (−0.987 − 0.159i)2-s + (−0.697 + 0.716i)3-s + (0.949 + 0.314i)4-s + (0.388 + 0.921i)5-s + (0.802 − 0.596i)6-s + (−0.437 + 0.899i)7-s + (−0.887 − 0.461i)8-s + (−0.0266 − 0.999i)9-s + (−0.237 − 0.971i)10-s + (0.977 + 0.211i)11-s + (−0.887 + 0.461i)12-s + (−0.437 − 0.899i)13-s + (0.574 − 0.818i)14-s + (−0.931 − 0.364i)15-s + (0.802 + 0.596i)16-s + (0.977 − 0.211i)17-s + ⋯
L(s)  = 1  + (−0.987 − 0.159i)2-s + (−0.697 + 0.716i)3-s + (0.949 + 0.314i)4-s + (0.388 + 0.921i)5-s + (0.802 − 0.596i)6-s + (−0.437 + 0.899i)7-s + (−0.887 − 0.461i)8-s + (−0.0266 − 0.999i)9-s + (−0.237 − 0.971i)10-s + (0.977 + 0.211i)11-s + (−0.887 + 0.461i)12-s + (−0.437 − 0.899i)13-s + (0.574 − 0.818i)14-s + (−0.931 − 0.364i)15-s + (0.802 + 0.596i)16-s + (0.977 − 0.211i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.503 - 0.864i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.503 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01565696340 + 0.02724523778i\)
\(L(\frac12)\) \(\approx\) \(0.01565696340 + 0.02724523778i\)
\(L(1)\) \(\approx\) \(0.4336872022 + 0.1627405916i\)
\(L(1)\) \(\approx\) \(0.4336872022 + 0.1627405916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.987 - 0.159i)T \)
3 \( 1 + (-0.697 + 0.716i)T \)
5 \( 1 + (0.388 + 0.921i)T \)
7 \( 1 + (-0.437 + 0.899i)T \)
11 \( 1 + (0.977 + 0.211i)T \)
13 \( 1 + (-0.437 - 0.899i)T \)
17 \( 1 + (0.977 - 0.211i)T \)
19 \( 1 + (-0.887 - 0.461i)T \)
23 \( 1 + (-0.998 + 0.0532i)T \)
29 \( 1 + (-0.132 - 0.991i)T \)
31 \( 1 + (-0.931 - 0.364i)T \)
37 \( 1 + (-0.437 + 0.899i)T \)
41 \( 1 + (-0.964 + 0.263i)T \)
43 \( 1 + (-0.833 - 0.552i)T \)
47 \( 1 + (-0.998 + 0.0532i)T \)
53 \( 1 + (-0.931 - 0.364i)T \)
59 \( 1 + (-0.887 - 0.461i)T \)
61 \( 1 + (0.910 + 0.413i)T \)
67 \( 1 + (-0.530 + 0.847i)T \)
71 \( 1 + (-0.237 + 0.971i)T \)
73 \( 1 + (0.994 - 0.106i)T \)
79 \( 1 + (0.185 - 0.982i)T \)
83 \( 1 + (-0.617 + 0.786i)T \)
89 \( 1 + (0.288 - 0.957i)T \)
97 \( 1 + (0.0797 - 0.996i)T \)
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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

−21.97674717399942467403520725475, −21.18530201870894970220005815357, −20.02303913809671613260264849135, −19.5981457365759758496794760406, −18.79617251501767962927742606073, −17.84371657621203702294460924192, −16.99852460728277801893173566208, −16.62765323441765580572624351184, −16.22848972206604344334649253756, −14.5166548138341359458451747292, −13.859083926991887522124692634626, −12.56839655109548376549836918399, −12.16260083403207681036102916684, −11.116256904763025816338416696983, −10.213989749406936133671127706261, −9.44262293892369358350125659047, −8.47760828290426927049583933459, −7.57974176682080076693864851283, −6.66237146242440547278488244698, −6.09585556767944383934061793550, −4.98825912296629918429180067914, −3.676064631534960120372244590485, −1.82212341406488832115551836838, −1.38207714620516977965930160187, −0.022594615847452008920714034362, 1.78062885588047121398422021493, 2.91815633967651598738920848471, 3.70808566289222282958548650164, 5.36193845213040122555349235912, 6.21288765320840837340901980676, 6.77319241769369286448801572418, 8.03177159549659299756635392130, 9.158671880643718551803082323283, 9.91774090346874589902249951778, 10.27013955274838299875304554451, 11.47684342521971409423928269686, 11.89276516598754170871119391424, 12.892190333181461213865658522738, 14.5964370562464895576100712479, 15.10985255593207337575324687017, 15.83651964781670515549033520352, 16.89782055947659385737778543747, 17.393593643185513894652347050419, 18.2403844788414196384224711150, 18.885627511341218272327563451029, 19.78673188551423795883862493009, 20.73543047047740948314475172130, 21.714788059947640573530520428466, 22.09681515900160331325713540393, 22.8258113012786625458798416286

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