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Properties

Label 1-709-709.79-r1-0-0
Degree $1$
Conductor $709$
Sign $-0.646 - 0.763i$
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  + (−0.995 − 0.0974i)2-s + (−0.468 − 0.883i)3-s + (0.981 + 0.194i)4-s + (0.999 + 0.0177i)5-s + (0.380 + 0.924i)6-s + (0.355 − 0.934i)7-s + (−0.957 − 0.288i)8-s + (−0.560 + 0.828i)9-s + (−0.993 − 0.115i)10-s + (−0.842 + 0.537i)11-s + (−0.288 − 0.957i)12-s + (0.159 − 0.987i)13-s + (−0.445 + 0.895i)14-s + (−0.453 − 0.891i)15-s + (0.924 + 0.380i)16-s + (0.999 + 0.0443i)17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0974i)2-s + (−0.468 − 0.883i)3-s + (0.981 + 0.194i)4-s + (0.999 + 0.0177i)5-s + (0.380 + 0.924i)6-s + (0.355 − 0.934i)7-s + (−0.957 − 0.288i)8-s + (−0.560 + 0.828i)9-s + (−0.993 − 0.115i)10-s + (−0.842 + 0.537i)11-s + (−0.288 − 0.957i)12-s + (0.159 − 0.987i)13-s + (−0.445 + 0.895i)14-s + (−0.453 − 0.891i)15-s + (0.924 + 0.380i)16-s + (0.999 + 0.0443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.646 - 0.763i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.646 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5039294293 - 1.086921900i\)
\(L(\frac12)\) \(\approx\) \(0.5039294293 - 1.086921900i\)
\(L(1)\) \(\approx\) \(0.6676549833 - 0.3416403482i\)
\(L(1)\) \(\approx\) \(0.6676549833 - 0.3416403482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.995 - 0.0974i)T \)
3 \( 1 + (-0.468 - 0.883i)T \)
5 \( 1 + (0.999 + 0.0177i)T \)
7 \( 1 + (0.355 - 0.934i)T \)
11 \( 1 + (-0.842 + 0.537i)T \)
13 \( 1 + (0.159 - 0.987i)T \)
17 \( 1 + (0.999 + 0.0443i)T \)
19 \( 1 + (0.684 + 0.728i)T \)
23 \( 1 + (-0.928 + 0.372i)T \)
29 \( 1 + (0.937 + 0.347i)T \)
31 \( 1 + (-0.838 - 0.545i)T \)
37 \( 1 + (-0.934 - 0.355i)T \)
41 \( 1 + (0.330 + 0.943i)T \)
43 \( 1 + (0.870 - 0.492i)T \)
47 \( 1 + (0.617 - 0.786i)T \)
53 \( 1 + (0.0532 - 0.998i)T \)
59 \( 1 + (0.288 - 0.957i)T \)
61 \( 1 + (-0.0886 - 0.996i)T \)
67 \( 1 + (-0.903 + 0.429i)T \)
71 \( 1 + (-0.993 + 0.115i)T \)
73 \( 1 + (0.971 - 0.237i)T \)
79 \( 1 + (0.476 + 0.879i)T \)
83 \( 1 + (-0.314 + 0.949i)T \)
89 \( 1 + (0.703 + 0.710i)T \)
97 \( 1 + (0.740 - 0.671i)T \)
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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

−22.42702444699246931495203973996, −21.51536458383135750466174330576, −21.18902317622502166411624852972, −20.52130631926489322021994395708, −19.23044260314512230149963260530, −18.33973190610941865173733316201, −17.87547905675606548891617177717, −17.03109662808335370595478510080, −16.1136542199376402375730769503, −15.78200083220216028830575739813, −14.60303941717497958613502931912, −13.93172952569064028332576201908, −12.32414549027505847586018715907, −11.691078449250342194765034207167, −10.678018411507598718713114399173, −10.14557078431853865034065332685, −9.08430198979086797434788388100, −8.8325931901475621345467157837, −7.50165251455930571509101808776, −6.16217058528170278453994691434, −5.72593620558961583761786455485, −4.827835430615936097095864617098, −3.1202416423722904857766287138, −2.27582033002266736696223254792, −1.01211493585038005378719665537, 0.4752972729021447534532964590, 1.3756065247073442268220920388, 2.177143515607764760865579437437, 3.34228487586646994422720142366, 5.26576333169080380421439053357, 5.851495904329996132957913447897, 6.97541695704566188615506748969, 7.69916190699418426278311712725, 8.26012087411337234537464976898, 9.773200870497329035849152852082, 10.31332629513266129020126678795, 10.961438169241714528499461165625, 12.156668549879299335376729936324, 12.80810430392353566438285534370, 13.76449196838429047956511206275, 14.57247212228029989382142836467, 15.95580199418767690736830338721, 16.71328553141695009712843888022, 17.50138694114116784136976807298, 17.99560375159592977425644213705, 18.48557153273446936277149878484, 19.61281071644549367911931667207, 20.45373650526319655328821672703, 20.93108045325950268553056560097, 22.10016927335439396668999865787

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