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Properties

Label 1-709-709.177-r0-0-0
Degree $1$
Conductor $709$
Sign $0.953 + 0.301i$
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  + (−0.999 + 0.0177i)2-s + (−0.996 − 0.0886i)3-s + (0.999 − 0.0354i)4-s + (0.842 + 0.537i)5-s + (0.997 + 0.0709i)6-s + (0.603 − 0.797i)7-s + (−0.998 + 0.0532i)8-s + (0.984 + 0.176i)9-s + (−0.852 − 0.522i)10-s + (0.781 + 0.624i)11-s + (−0.998 − 0.0532i)12-s + (0.388 − 0.921i)13-s + (−0.589 + 0.807i)14-s + (−0.792 − 0.610i)15-s + (0.997 − 0.0709i)16-s + (0.150 + 0.988i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0177i)2-s + (−0.996 − 0.0886i)3-s + (0.999 − 0.0354i)4-s + (0.842 + 0.537i)5-s + (0.997 + 0.0709i)6-s + (0.603 − 0.797i)7-s + (−0.998 + 0.0532i)8-s + (0.984 + 0.176i)9-s + (−0.852 − 0.522i)10-s + (0.781 + 0.624i)11-s + (−0.998 − 0.0532i)12-s + (0.388 − 0.921i)13-s + (−0.589 + 0.807i)14-s + (−0.792 − 0.610i)15-s + (0.997 − 0.0709i)16-s + (0.150 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.953 + 0.301i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.953 + 0.301i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9560205756 + 0.1475772915i\)
\(L(\frac12)\) \(\approx\) \(0.9560205756 + 0.1475772915i\)
\(L(1)\) \(\approx\) \(0.7515617253 + 0.04414127411i\)
\(L(1)\) \(\approx\) \(0.7515617253 + 0.04414127411i\)

Euler product

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

−22.30131314961349328192067340033, −21.54266998818900180595021133530, −21.11946608573928630000078978851, −20.17750167756760039254750242476, −18.9318075853147447699531336167, −18.372039424740103081234505671322, −17.661886574225243519830021309819, −16.94669273534947965876110240974, −16.331160117041616311967059546881, −15.602098310250744937778376978105, −14.429497561743167872475461722512, −13.38123489113136975410855321955, −12.104768031161053123990839484963, −11.67039721987746437408684227881, −10.92725823498004134584676907410, −9.84824223433965377717885321012, −9.06165809614748611840611129281, −8.57117862064968520815589987398, −7.0082274534039692880936469864, −6.42806403084544822555538815420, −5.44833111541990134288907370823, −4.719139121937442106478655591054, −2.95452658181469491087164146647, −1.66569591052798539523895323917, −0.95933122649861886333464674381, 1.15848546830640044479341867880, 1.65728505500757303281326601931, 3.20794623571047188766527090897, 4.56922624782579572936371415785, 5.88992840679423327868335931861, 6.343690856238462434080404551113, 7.376339198909171086971816973, 8.038654336631106451540289938333, 9.53597011990262477178467907436, 10.121272599841034605571595525333, 10.80562954259425777993087879529, 11.47237092460942673599233604031, 12.48478449045287243298660554589, 13.46749039716069411844596250648, 14.67583240976594290348343996198, 15.32585324617089147752673472493, 16.59323371655073878137065288480, 17.17729955834157358615811926277, 17.65641700818415945950277071878, 18.261687745227605199281619109, 19.18888272304226468184070417129, 20.16804633960725034079998306010, 21.06488198595653570633873297497, 21.64016204859232820519070607323, 22.91744053327862307775448174601

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