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Properties

Label 1-709-709.604-r0-0-0
Degree $1$
Conductor $709$
Sign $0.813 + 0.581i$
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  + (−0.132 + 0.991i)2-s + (−0.617 − 0.786i)3-s + (−0.964 − 0.263i)4-s + (−0.437 + 0.899i)5-s + (0.861 − 0.507i)6-s + (0.802 + 0.596i)7-s + (0.388 − 0.921i)8-s + (−0.237 + 0.971i)9-s + (−0.833 − 0.552i)10-s + (−0.339 − 0.940i)11-s + (0.388 + 0.921i)12-s + (0.802 − 0.596i)13-s + (−0.697 + 0.716i)14-s + (0.977 − 0.211i)15-s + (0.861 + 0.507i)16-s + (−0.339 + 0.940i)17-s + ⋯
L(s)  = 1  + (−0.132 + 0.991i)2-s + (−0.617 − 0.786i)3-s + (−0.964 − 0.263i)4-s + (−0.437 + 0.899i)5-s + (0.861 − 0.507i)6-s + (0.802 + 0.596i)7-s + (0.388 − 0.921i)8-s + (−0.237 + 0.971i)9-s + (−0.833 − 0.552i)10-s + (−0.339 − 0.940i)11-s + (0.388 + 0.921i)12-s + (0.802 − 0.596i)13-s + (−0.697 + 0.716i)14-s + (0.977 − 0.211i)15-s + (0.861 + 0.507i)16-s + (−0.339 + 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8352274879 + 0.2676648679i\)
\(L(\frac12)\) \(\approx\) \(0.8352274879 + 0.2676648679i\)
\(L(1)\) \(\approx\) \(0.7211543728 + 0.2378199226i\)
\(L(1)\) \(\approx\) \(0.7211543728 + 0.2378199226i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.132 + 0.991i)T \)
3 \( 1 + (-0.617 - 0.786i)T \)
5 \( 1 + (-0.437 + 0.899i)T \)
7 \( 1 + (0.802 + 0.596i)T \)
11 \( 1 + (-0.339 - 0.940i)T \)
13 \( 1 + (0.802 - 0.596i)T \)
17 \( 1 + (-0.339 + 0.940i)T \)
19 \( 1 + (0.388 - 0.921i)T \)
23 \( 1 + (-0.887 - 0.461i)T \)
29 \( 1 + (-0.931 + 0.364i)T \)
31 \( 1 + (0.977 - 0.211i)T \)
37 \( 1 + (0.802 + 0.596i)T \)
41 \( 1 + (0.734 - 0.678i)T \)
43 \( 1 + (-0.530 - 0.847i)T \)
47 \( 1 + (-0.887 - 0.461i)T \)
53 \( 1 + (0.977 - 0.211i)T \)
59 \( 1 + (0.388 - 0.921i)T \)
61 \( 1 + (-0.769 + 0.638i)T \)
67 \( 1 + (0.949 - 0.314i)T \)
71 \( 1 + (-0.833 + 0.552i)T \)
73 \( 1 + (0.574 + 0.818i)T \)
79 \( 1 + (0.994 - 0.106i)T \)
83 \( 1 + (0.288 - 0.957i)T \)
89 \( 1 + (0.484 - 0.874i)T \)
97 \( 1 + (0.658 + 0.752i)T \)
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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

−22.59876707073482480456794599185, −21.30458536249923696381514752786, −20.898977949487124308158045953040, −20.363037942440422129719341246833, −19.63956794342579582550626046146, −18.13309941302088272304160899301, −17.90978736898397523321612529539, −16.76218705214614351765542561574, −16.27743450733043214232940381546, −15.18453235851436965568104576897, −14.13956943916717666204647898348, −13.257635879349037216168744964752, −12.21572979065637388690162576819, −11.579847292507368777027180809303, −11.02937705503919239783267662092, −9.8998489131056294830423065157, −9.41129375161887465339706159035, −8.31268050358241864333064459428, −7.51587016273961009522968359021, −5.82262988208800482161511394827, −4.74350431406480836587910299706, −4.386120727380137730608267056219, −3.53199724492545532165142422793, −1.875599610155288686877593906295, −0.86865025632871905771970563856, 0.71131219045601758930532815689, 2.20160303018213305334956510911, 3.57186891234996415234761760735, 4.85832495321050705904055054173, 5.86604187137716034377409568913, 6.299161224448859051943737284936, 7.38004389157478898244134418408, 8.15658476057249592000643288929, 8.62816844459971234757982460139, 10.30882309798506000665815219438, 11.03332345854897427194169609036, 11.76633292805995380424540340466, 12.990962132528653745711908615403, 13.67212618119741130786650096618, 14.54566724628598439959560553688, 15.41336043121084356587837255028, 16.03138227594841216655497930592, 17.12360670831503246281186502093, 17.92044456825323871377016558130, 18.41477295349799771790145960252, 18.95092658324765853740308484836, 19.92927894631183927673374490286, 21.528971409347000689350120762, 22.1482159019307445582055959246, 22.82733419796402230159163218027

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