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Properties

Label 1-709-709.24-r1-0-0
Degree $1$
Conductor $709$
Sign $0.855 - 0.517i$
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  + (0.0177 + 0.999i)2-s + (−0.996 − 0.0886i)3-s + (−0.999 + 0.0354i)4-s + (−0.842 − 0.537i)5-s + (0.0709 − 0.997i)6-s + (0.603 − 0.797i)7-s + (−0.0532 − 0.998i)8-s + (0.984 + 0.176i)9-s + (0.522 − 0.852i)10-s + (−0.781 − 0.624i)11-s + (0.998 + 0.0532i)12-s + (−0.921 − 0.388i)13-s + (0.807 + 0.589i)14-s + (0.792 + 0.610i)15-s + (0.997 − 0.0709i)16-s + (0.988 − 0.150i)17-s + ⋯
L(s)  = 1  + (0.0177 + 0.999i)2-s + (−0.996 − 0.0886i)3-s + (−0.999 + 0.0354i)4-s + (−0.842 − 0.537i)5-s + (0.0709 − 0.997i)6-s + (0.603 − 0.797i)7-s + (−0.0532 − 0.998i)8-s + (0.984 + 0.176i)9-s + (0.522 − 0.852i)10-s + (−0.781 − 0.624i)11-s + (0.998 + 0.0532i)12-s + (−0.921 − 0.388i)13-s + (0.807 + 0.589i)14-s + (0.792 + 0.610i)15-s + (0.997 − 0.0709i)16-s + (0.988 − 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.855 - 0.517i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.855 - 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7990003235 - 0.2228058313i\)
\(L(\frac12)\) \(\approx\) \(0.7990003235 - 0.2228058313i\)
\(L(1)\) \(\approx\) \(0.6260580126 + 0.1158563203i\)
\(L(1)\) \(\approx\) \(0.6260580126 + 0.1158563203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.0177 + 0.999i)T \)
3 \( 1 + (-0.996 - 0.0886i)T \)
5 \( 1 + (-0.842 - 0.537i)T \)
7 \( 1 + (0.603 - 0.797i)T \)
11 \( 1 + (-0.781 - 0.624i)T \)
13 \( 1 + (-0.921 - 0.388i)T \)
17 \( 1 + (0.988 - 0.150i)T \)
19 \( 1 + (0.545 + 0.838i)T \)
23 \( 1 + (-0.347 + 0.937i)T \)
29 \( 1 + (0.355 - 0.934i)T \)
31 \( 1 + (0.380 + 0.924i)T \)
37 \( 1 + (0.797 + 0.603i)T \)
41 \( 1 + (-0.979 - 0.202i)T \)
43 \( 1 + (0.722 - 0.691i)T \)
47 \( 1 + (0.769 - 0.638i)T \)
53 \( 1 + (0.991 + 0.132i)T \)
59 \( 1 + (-0.998 + 0.0532i)T \)
61 \( 1 + (-0.297 + 0.954i)T \)
67 \( 1 + (-0.0620 - 0.998i)T \)
71 \( 1 + (0.522 + 0.852i)T \)
73 \( 1 + (-0.982 - 0.185i)T \)
79 \( 1 + (0.194 + 0.981i)T \)
83 \( 1 + (-0.716 + 0.697i)T \)
89 \( 1 + (0.141 - 0.989i)T \)
97 \( 1 + (0.999 - 0.00887i)T \)
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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

−22.252330635999850916307380479020, −21.84152438293537210351867469374, −20.973301457158262862017658788390, −20.08885555385626419919619119004, −19.05862762805023829320904522803, −18.42426088636457508545650409426, −17.93786982144110795320687215574, −16.96676984523076119844738377567, −15.89413172120212833188134580919, −14.99144512575317510433079018246, −14.336333767921209920500723782701, −12.880999941192444514630088333616, −12.16921031620754466040707670707, −11.74887376560231986066516076584, −10.89179044919172880838136071884, −10.19740241352570041644937122156, −9.287414023239771605475350335502, −8.003267504587406948111985776971, −7.28487907304753761289206986442, −5.89099363452997625886439760612, −4.85872558405360044695171972203, −4.43301661358034605729370474807, −3.00863029734128661244177243622, −2.12460038309529556392022514529, −0.71152428950932249785325315837, 0.397785641246457703020196062253, 1.15451773710923592194627356129, 3.46506714699294840598599090977, 4.42939418536845931086249542385, 5.20984321160235236313742907643, 5.79025153713263510291063714661, 7.257426156789142412556703845230, 7.631849883680228959819431784902, 8.358588234641789059423294924402, 9.85922287947353040754976268369, 10.45936168986970006261916624360, 11.737109881026974663220432537655, 12.286265493451776812898250916392, 13.33484559807127255825246621271, 14.08315616002468662630261731522, 15.272739530978172564538930332876, 15.86153940550624803895532713497, 16.80635547330982372412668024184, 17.02344342519973412157812114472, 18.097520863301210060158744497277, 18.78826606320733552104803997871, 19.75000839285143777601929587592, 20.9354757980141103994355832666, 21.6762548785494901083528144054, 22.811261147810550596376288360494

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