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Properties

Label 1-799-799.610-r1-0-0
Degree $1$
Conductor $799$
Sign $-0.0758 - 0.997i$
Analytic cond. $85.8644$
Root an. cond. $85.8644$
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  + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s − i·8-s − i·9-s + (−0.707 − 0.707i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + 13-s + (−0.707 + 0.707i)14-s − i·15-s + 16-s + ⋯
L(s)  = 1  + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s − i·8-s − i·9-s + (−0.707 − 0.707i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + 13-s + (−0.707 + 0.707i)14-s − i·15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.0758 - 0.997i$
Analytic conductor: \(85.8644\)
Root analytic conductor: \(85.8644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (610, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 799,\ (1:\ ),\ -0.0758 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5960032172 + 0.6430858675i\)
\(L(\frac12)\) \(\approx\) \(-0.5960032172 + 0.6430858675i\)
\(L(1)\) \(\approx\) \(0.4020816796 + 0.6381771585i\)
\(L(1)\) \(\approx\) \(0.4020816796 + 0.6381771585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
47 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + iT \)
53 \( 1 + iT \)
59 \( 1 - iT \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + iT \)
89 \( 1 - T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−21.19871245096746904828942535080, −20.73319456958272510029734830829, −19.824739479103169772563467759, −19.0838789638717632428006464322, −18.50263533176501143863037154539, −17.48612206585924578525427772567, −16.865608379327863668618774343117, −16.19021331556825748328085753590, −14.692044057823082237288274819760, −13.752574716769626673277533918667, −13.19207254877098716085848846710, −12.22687700161990462546503426791, −11.620069779184299330371631242307, −11.02157929841683593111470102877, −10.258348349592576069456014989818, −8.73319477534947819167091715709, −8.32204752004640176920458777070, −7.33158166778792463748095831044, −6.05351427025237373459125068560, −5.08287573619291116558008222546, −4.21036025809992843438271751390, −3.42526265396799955543567489565, −1.71844383878501173279357402839, −1.10271051938707929271928320125, −0.279795632355618693488424861683, 1.21597973596880866619780360400, 3.15871165281294742882479435897, 4.12733605125074982397939336337, 4.8377272773937066191748082558, 5.781008629056839800521423929388, 6.6512121473342045059284251622, 7.37267580776225665911016530346, 8.5690895761467750052591769868, 9.1824579616050164584914412587, 10.242909797874968791738708959372, 11.28317785831267864568871576252, 11.765763637724753793658888791867, 12.88206102171995325027990627002, 14.106833503070074327198340210885, 14.99100197415405702026912435004, 15.34392088576912672300143569577, 15.96338966700436795868245064905, 17.07197781726204353304104692383, 17.669091644002232460920636090689, 18.37890061944628235972553547332, 19.136558519229769227143762489887, 20.39701138855840770920158116416, 21.37719293445514728339057756747, 22.21552170575680315849459406580, 22.60414914469051867406513269477

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