Properties

Label 1-731-731.394-r0-0-0
Degree $1$
Conductor $731$
Sign $0.968 - 0.248i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + (−0.707 + 0.707i)2-s + (0.793 − 0.608i)3-s i·4-s + (0.608 + 0.793i)5-s + (−0.130 + 0.991i)6-s + (0.608 − 0.793i)7-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (−0.991 − 0.130i)10-s + (−0.923 + 0.382i)11-s + (−0.608 − 0.793i)12-s + (0.866 − 0.5i)13-s + (0.130 + 0.991i)14-s + (0.965 + 0.258i)15-s − 16-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.793 − 0.608i)3-s i·4-s + (0.608 + 0.793i)5-s + (−0.130 + 0.991i)6-s + (0.608 − 0.793i)7-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (−0.991 − 0.130i)10-s + (−0.923 + 0.382i)11-s + (−0.608 − 0.793i)12-s + (0.866 − 0.5i)13-s + (0.130 + 0.991i)14-s + (0.965 + 0.258i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.566912255 - 0.1977912302i\)
\(L(\frac12)\) \(\approx\) \(1.566912255 - 0.1977912302i\)
\(L(1)\) \(\approx\) \(1.160433746 + 0.02668861197i\)
\(L(1)\) \(\approx\) \(1.160433746 + 0.02668861197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.793 - 0.608i)T \)
5 \( 1 + (0.608 + 0.793i)T \)
7 \( 1 + (0.608 - 0.793i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (-0.130 + 0.991i)T \)
29 \( 1 + (0.991 - 0.130i)T \)
31 \( 1 + (0.793 - 0.608i)T \)
37 \( 1 + (0.793 - 0.608i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.608 + 0.793i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.130 + 0.991i)T \)
73 \( 1 + (-0.991 + 0.130i)T \)
79 \( 1 + (0.793 + 0.608i)T \)
83 \( 1 + (-0.258 - 0.965i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−21.94015156387242899589437216029, −21.38356991087055210998226847951, −20.90205747779674534443190725617, −20.40100454455255987155334415147, −19.35299296833443995594750961965, −18.47970384957246386054901157256, −18.01769650930888561909710166895, −16.69589277566894729080720585659, −16.25761805343553171186463349777, −15.37317116336914015145225602008, −14.15923220366533957080903420937, −13.442337065271967565909169315562, −12.60192777510794309109445158481, −11.678579355979957701516216639308, −10.5968377227784910385441618234, −10.010720654684273650782348780118, −9.02507622258589876888081029774, −8.32817657460992534666077709354, −8.11765828913136428601541593793, −6.323330263463694975714142073365, −5.05380471882769913460062919446, −4.30582314296816074357349128145, −3.048343034464455792810768628307, −2.21731647968357332491168434479, −1.346883475423141167160938864120, 0.96238206639914401132265737025, 2.00737122536275898572392619481, 2.951019122783503595961351222870, 4.39902546906002872789878384096, 5.682903202053309211743258851546, 6.55769885858542089497292489465, 7.42990907882066892016417273785, 7.89022190529639017846560030944, 8.88064433994521436983474340396, 9.8632589048613257982709111616, 10.57038589048988080889032904350, 11.36516003010888664334562919396, 13.04171974939850277796451079851, 13.715320615647445826446605822073, 14.19808237932833102691192924948, 15.29030342221576261033140755924, 15.61604847241092616246213032735, 17.18677051646532824244871004269, 17.769762213551820479557449698717, 18.22504440395636629248391337629, 19.07088084940464647827024418273, 19.91533567100771208056359924142, 20.668640626073568254229785515869, 21.45288978862418284854861318071, 22.96473539599554848192150449607

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