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Properties

Label 1-7e3-343.45-r1-0-0
Degree $1$
Conductor $343$
Sign $0.393 + 0.919i$
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  + (−0.961 + 0.274i)2-s + (0.788 + 0.615i)3-s + (0.849 − 0.527i)4-s + (0.934 + 0.355i)5-s + (−0.926 − 0.375i)6-s + (−0.672 + 0.740i)8-s + (0.243 + 0.969i)9-s + (−0.996 − 0.0853i)10-s + (0.910 − 0.414i)11-s + (0.994 + 0.106i)12-s + (0.949 + 0.315i)13-s + (0.518 + 0.855i)15-s + (0.443 − 0.896i)16-s + (−0.656 − 0.754i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.961 + 0.274i)2-s + (0.788 + 0.615i)3-s + (0.849 − 0.527i)4-s + (0.934 + 0.355i)5-s + (−0.926 − 0.375i)6-s + (−0.672 + 0.740i)8-s + (0.243 + 0.969i)9-s + (−0.996 − 0.0853i)10-s + (0.910 − 0.414i)11-s + (0.994 + 0.106i)12-s + (0.949 + 0.315i)13-s + (0.518 + 0.855i)15-s + (0.443 − 0.896i)16-s + (−0.656 − 0.754i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $0.393 + 0.919i$
Analytic conductor: \(36.8604\)
Root analytic conductor: \(36.8604\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 343,\ (1:\ ),\ 0.393 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.995019886 + 1.316026787i\)
\(L(\frac12)\) \(\approx\) \(1.995019886 + 1.316026787i\)
\(L(1)\) \(\approx\) \(1.181509274 + 0.4592297215i\)
\(L(1)\) \(\approx\) \(1.181509274 + 0.4592297215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.961 + 0.274i)T \)
3 \( 1 + (0.788 + 0.615i)T \)
5 \( 1 + (0.934 + 0.355i)T \)
11 \( 1 + (0.910 - 0.414i)T \)
13 \( 1 + (0.949 + 0.315i)T \)
17 \( 1 + (-0.656 - 0.754i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.263 + 0.964i)T \)
29 \( 1 + (0.967 + 0.253i)T \)
31 \( 1 + (0.988 + 0.149i)T \)
37 \( 1 + (0.325 - 0.945i)T \)
41 \( 1 + (-0.284 + 0.958i)T \)
43 \( 1 + (-0.672 - 0.740i)T \)
47 \( 1 + (-0.481 - 0.876i)T \)
53 \( 1 + (0.942 + 0.335i)T \)
59 \( 1 + (-0.687 + 0.725i)T \)
61 \( 1 + (-0.117 - 0.993i)T \)
67 \( 1 + (0.365 - 0.930i)T \)
71 \( 1 + (0.967 - 0.253i)T \)
73 \( 1 + (-0.481 + 0.876i)T \)
79 \( 1 + (0.826 - 0.563i)T \)
83 \( 1 + (-0.991 - 0.127i)T \)
89 \( 1 + (0.640 + 0.768i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−24.824551421322189357700032282920, −24.160429296232166253890401408036, −22.66058015165690427481254368211, −21.46396874487344015485854672818, −20.64836809067744385797428369051, −20.10587614216781156893131480160, −19.16119982271541644897489847025, −18.23057216549560318071148349161, −17.63544592014216929517776356670, −16.75818307439609652762972193538, −15.60743715314336744262535824613, −14.524849506920681378306323273571, −13.52521062386250474090236428644, −12.63379741656504679763645775275, −11.80966074570890455172587291048, −10.37403017716611359820311178957, −9.63261796615847673980680748132, −8.64059619364616467193891425294, −8.1482956835046151212671212475, −6.653503502970249457187714996488, −6.18146802645845893573545127683, −4.09934632802299921735396613224, −2.83030989534358762193721773315, −1.73225891724580407794125567346, −1.02614053435420998251316569958, 1.2038485642096065299221849432, 2.35400813348445916475185447821, 3.401614302414946206335758353085, 5.01648564213272152449909564347, 6.26182415828377882008063482803, 7.09861439833952209933305032078, 8.45502036975368599717880714103, 9.161031521132727392192634078602, 9.78329648349737504890566043529, 10.824644801471387921997968292907, 11.60588090536560489500994984087, 13.62062927062370521445946707545, 13.96218337470450182650111842982, 15.12898306101106306380575298853, 15.908894170649196711185769725255, 16.76890556572627880876500636020, 17.78531837745076897102097093594, 18.518827920844548116340522468327, 19.61096955457196165021277996746, 20.17576638293351681808274324903, 21.31725963214839283482312344135, 21.789629051877760061087903584559, 23.11235895187428399649971438309, 24.579619014890128238226633679655, 24.99279114812174759159281849000

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