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Properties

Label 1-7e3-343.167-r1-0-0
Degree $1$
Conductor $343$
Sign $0.946 + 0.323i$
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  + (−0.345 + 0.938i)2-s + (0.572 − 0.820i)3-s + (−0.761 − 0.648i)4-s + (−0.926 − 0.375i)5-s + (0.572 + 0.820i)6-s + (0.871 − 0.490i)8-s + (−0.345 − 0.938i)9-s + (0.672 − 0.740i)10-s + (0.518 + 0.855i)11-s + (−0.967 + 0.253i)12-s + (−0.718 + 0.695i)13-s + (−0.838 + 0.545i)15-s + (0.159 + 0.987i)16-s + (0.462 + 0.886i)17-s + 18-s − 19-s + ⋯
L(s)  = 1  + (−0.345 + 0.938i)2-s + (0.572 − 0.820i)3-s + (−0.761 − 0.648i)4-s + (−0.926 − 0.375i)5-s + (0.572 + 0.820i)6-s + (0.871 − 0.490i)8-s + (−0.345 − 0.938i)9-s + (0.672 − 0.740i)10-s + (0.518 + 0.855i)11-s + (−0.967 + 0.253i)12-s + (−0.718 + 0.695i)13-s + (−0.838 + 0.545i)15-s + (0.159 + 0.987i)16-s + (0.462 + 0.886i)17-s + 18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(36.8604\)
Root analytic conductor: \(36.8604\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 343,\ (1:\ ),\ 0.946 + 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328213484 + 0.2209815735i\)
\(L(\frac12)\) \(\approx\) \(1.328213484 + 0.2209815735i\)
\(L(1)\) \(\approx\) \(0.8756084133 + 0.1128922868i\)
\(L(1)\) \(\approx\) \(0.8756084133 + 0.1128922868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.345 + 0.938i)T \)
3 \( 1 + (0.572 - 0.820i)T \)
5 \( 1 + (-0.926 - 0.375i)T \)
11 \( 1 + (0.518 + 0.855i)T \)
13 \( 1 + (-0.718 + 0.695i)T \)
17 \( 1 + (0.462 + 0.886i)T \)
19 \( 1 - T \)
23 \( 1 + (0.801 - 0.598i)T \)
29 \( 1 + (0.801 + 0.598i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + (-0.462 - 0.886i)T \)
41 \( 1 + (0.997 - 0.0640i)T \)
43 \( 1 + (0.871 + 0.490i)T \)
47 \( 1 + (0.838 + 0.545i)T \)
53 \( 1 + (-0.981 + 0.191i)T \)
59 \( 1 + (0.997 + 0.0640i)T \)
61 \( 1 + (-0.0320 - 0.999i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (0.801 - 0.598i)T \)
73 \( 1 + (0.838 - 0.545i)T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + (0.949 + 0.315i)T \)
89 \( 1 + (0.0960 - 0.995i)T \)
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

−24.90347727593968259561740755991, −23.41788774353495501043855777345, −22.55667755455872074829985684651, −21.86376638287560292992199167302, −21.00261983020150647805146899771, −20.13311882088802556281611805546, −19.30755431555852921374189802562, −18.97174112221673771454231993296, −17.51914507137059309454696104258, −16.58833944660315811003893966447, −15.65404960172297151903548527566, −14.61485704607731305144313844384, −13.84934223214580142864925763904, −12.60097854794550633701637678520, −11.570092221308766978167406325486, −10.84177906525767861411498168694, −10.00121299564186197776393013758, −8.93931485519534998359505943509, −8.20473943861477332799560435378, −7.22917218331127407788970497935, −5.24109821257265656793372075677, −4.18511206429581755627557131670, −3.301048557467932380884220215725, −2.57555552117068203623499908994, −0.66771720612422866948741159915, 0.726172507595641326274611346000, 2.00779290475673040849643633817, 3.83914102095075404998048675947, 4.71438885586144777997227862617, 6.25762336468454767330504246665, 7.12520733461068686566975461143, 7.808855825558321935741342380753, 8.763765450619179234702696712132, 9.44861622179473314135521991919, 10.92891442986101385670836944558, 12.46358623310539817907996876128, 12.71352662800172641198971548771, 14.3305944038857719126313413509, 14.695888835679842847226110173223, 15.62600651421497639288686345426, 16.85363810687298815604663289837, 17.38910391139702149524240652145, 18.6126799882608670379541274005, 19.36618291907924970768926945403, 19.77642163292959773309354939741, 21.03546252646591238180663372385, 22.51899475249887276706720760116, 23.38795602535491658176605716008, 23.9468557596239900383966435971, 24.70569520528619811168222893130

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