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Properties

Label 1-7e3-343.5-r1-0-0
Degree $1$
Conductor $343$
Sign $-0.908 + 0.417i$
Analytic cond. $36.8604$
Root an. cond. $36.8604$
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  + (0.656 + 0.754i)2-s + (0.814 + 0.580i)3-s + (−0.138 + 0.990i)4-s + (0.640 − 0.768i)5-s + (0.0960 + 0.995i)6-s + (−0.838 + 0.545i)8-s + (0.325 + 0.945i)9-s + (0.999 − 0.0213i)10-s + (−0.994 − 0.106i)11-s + (−0.687 + 0.725i)12-s + (0.761 + 0.648i)13-s + (0.967 − 0.253i)15-s + (−0.961 − 0.274i)16-s + (−0.977 + 0.212i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.656 + 0.754i)2-s + (0.814 + 0.580i)3-s + (−0.138 + 0.990i)4-s + (0.640 − 0.768i)5-s + (0.0960 + 0.995i)6-s + (−0.838 + 0.545i)8-s + (0.325 + 0.945i)9-s + (0.999 − 0.0213i)10-s + (−0.994 − 0.106i)11-s + (−0.687 + 0.725i)12-s + (0.761 + 0.648i)13-s + (0.967 − 0.253i)15-s + (−0.961 − 0.274i)16-s + (−0.977 + 0.212i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $-0.908 + 0.417i$
Analytic conductor: \(36.8604\)
Root analytic conductor: \(36.8604\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 343,\ (1:\ ),\ -0.908 + 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7553843577 + 3.455128415i\)
\(L(\frac12)\) \(\approx\) \(0.7553843577 + 3.455128415i\)
\(L(1)\) \(\approx\) \(1.428989509 + 1.336316132i\)
\(L(1)\) \(\approx\) \(1.428989509 + 1.336316132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.656 + 0.754i)T \)
3 \( 1 + (0.814 + 0.580i)T \)
5 \( 1 + (0.640 - 0.768i)T \)
11 \( 1 + (-0.994 - 0.106i)T \)
13 \( 1 + (0.761 + 0.648i)T \)
17 \( 1 + (-0.977 + 0.212i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.443 + 0.896i)T \)
29 \( 1 + (-0.997 + 0.0640i)T \)
31 \( 1 + (0.733 + 0.680i)T \)
37 \( 1 + (-0.304 + 0.952i)T \)
41 \( 1 + (0.949 + 0.315i)T \)
43 \( 1 + (-0.838 - 0.545i)T \)
47 \( 1 + (0.263 + 0.964i)T \)
53 \( 1 + (0.996 - 0.0853i)T \)
59 \( 1 + (-0.201 + 0.979i)T \)
61 \( 1 + (0.934 - 0.355i)T \)
67 \( 1 + (0.955 + 0.294i)T \)
71 \( 1 + (-0.997 - 0.0640i)T \)
73 \( 1 + (0.263 - 0.964i)T \)
79 \( 1 + (-0.988 - 0.149i)T \)
83 \( 1 + (-0.0320 - 0.999i)T \)
89 \( 1 + (0.536 - 0.843i)T \)
97 \( 1 + (0.222 - 0.974i)T \)
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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

−24.35940392284728598865011029593, −23.23244629876627841108006265839, −22.55772398890176925174813652298, −21.49885202878751147378102668131, −20.711052156816875951758353653055, −20.07983162824584689390597821002, −18.96811476101275760339827815070, −18.296217083629584329025937701267, −17.72809135747039924853617532658, −15.66363598891128152156205997979, −15.05836205502415251282433551284, −14.08318892377722497179915333251, −13.23383193329342624163830036633, −12.94866780436338565990750964872, −11.426812875835776174117113807138, −10.61992669068715490458735333142, −9.6508694014202793431813768277, −8.66186045038328418067512701716, −7.26741047347424971380013549525, −6.34140788706790489732525539058, −5.266943341312043342627790146390, −3.80068170420995081720398695563, −2.70398577969711146035942878712, −2.21138955531206054685460687564, −0.68149959699666687706897823902, 1.82036379397360481940023270243, 3.09172914363369660957486173842, 4.20301176407140702314176179606, 5.08838704403048872915540653211, 5.99347919357576957614478710238, 7.38702703467024780589209966352, 8.42936758443637833696599039117, 9.01298830996999179199044044379, 10.12276353082679676253946355852, 11.45283323533248485421164866768, 12.8758019570946771822132348339, 13.46228067255665278630601522031, 14.09749060888364096648418794111, 15.27017856506386224295898359867, 15.964535331943607257902041026078, 16.63820354721244184274826543220, 17.70568666950197452980212585581, 18.761510512861837037998589136906, 20.17863431291804212946691494477, 20.939986438063674549570244545367, 21.3711636486661116050800677409, 22.31193851886409679518706820363, 23.47432774921319257226689302730, 24.32470806586646824253974157865, 25.040995234106950830843630797588

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