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Properties

Label 1-7e3-343.162-r0-0-0
Degree $1$
Conductor $343$
Sign $-0.715 - 0.698i$
Analytic cond. $1.59288$
Root an. cond. $1.59288$
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  + (−0.572 − 0.820i)2-s + (−0.462 − 0.886i)3-s + (−0.345 + 0.938i)4-s + (−0.981 − 0.191i)5-s + (−0.462 + 0.886i)6-s + (0.967 − 0.253i)8-s + (−0.572 + 0.820i)9-s + (0.404 + 0.914i)10-s + (0.871 + 0.490i)11-s + (0.991 − 0.127i)12-s + (0.926 − 0.375i)13-s + (0.284 + 0.958i)15-s + (−0.761 − 0.648i)16-s + (0.518 − 0.855i)17-s + 18-s + 19-s + ⋯
L(s)  = 1  + (−0.572 − 0.820i)2-s + (−0.462 − 0.886i)3-s + (−0.345 + 0.938i)4-s + (−0.981 − 0.191i)5-s + (−0.462 + 0.886i)6-s + (0.967 − 0.253i)8-s + (−0.572 + 0.820i)9-s + (0.404 + 0.914i)10-s + (0.871 + 0.490i)11-s + (0.991 − 0.127i)12-s + (0.926 − 0.375i)13-s + (0.284 + 0.958i)15-s + (−0.761 − 0.648i)16-s + (0.518 − 0.855i)17-s + 18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $-0.715 - 0.698i$
Analytic conductor: \(1.59288\)
Root analytic conductor: \(1.59288\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 343,\ (0:\ ),\ -0.715 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2380486384 - 0.5841358796i\)
\(L(\frac12)\) \(\approx\) \(0.2380486384 - 0.5841358796i\)
\(L(1)\) \(\approx\) \(0.4829460470 - 0.3936831621i\)
\(L(1)\) \(\approx\) \(0.4829460470 - 0.3936831621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.572 - 0.820i)T \)
3 \( 1 + (-0.462 - 0.886i)T \)
5 \( 1 + (-0.981 - 0.191i)T \)
11 \( 1 + (0.871 + 0.490i)T \)
13 \( 1 + (0.926 - 0.375i)T \)
17 \( 1 + (0.518 - 0.855i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.949 + 0.315i)T \)
29 \( 1 + (-0.949 - 0.315i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (0.518 - 0.855i)T \)
41 \( 1 + (0.0320 + 0.999i)T \)
43 \( 1 + (0.967 + 0.253i)T \)
47 \( 1 + (0.284 - 0.958i)T \)
53 \( 1 + (-0.0960 - 0.995i)T \)
59 \( 1 + (0.0320 - 0.999i)T \)
61 \( 1 + (0.718 + 0.695i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.949 + 0.315i)T \)
73 \( 1 + (0.284 + 0.958i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (0.159 - 0.987i)T \)
89 \( 1 + (-0.672 - 0.740i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−25.50858262720835807699333828565, −24.01160875725236703678666284826, −23.760519465101634388468560875239, −22.56703941762832520873354440241, −22.12614644019526823607745381942, −20.65705965254869727619652780349, −19.81548563463944352335469279416, −18.86107436870574258690060843270, −18.02358174845733646836820794589, −16.84103677272265525845432061457, −16.3042666385536809715332565991, −15.59732486446468539741098567917, −14.69853435576006727160924961560, −13.96662087281894346565296341137, −12.20965512299351412636430553686, −11.21365209788416671742592605640, −10.576925769477540711476371030143, −9.34764708485309703776904909931, −8.628366822824843102329839524309, −7.56194071366858765914576135902, −6.37024804205179369379528703684, −5.60662728420068706733003320562, −4.24015480689897196839315041620, −3.566621385359121755734725089040, −1.12436641866668553546269333254, 0.65892816302553507846094943347, 1.73400200648642337686382875936, 3.21620790264627415777872400894, 4.22530874986589612187501841121, 5.65164979642055723267731977601, 7.19859241652675917008239639180, 7.70733370470121616102253139230, 8.75208242080405817615907846714, 9.83654662466327929099045308094, 11.30934056839275305448813296453, 11.55234162939437160181695240455, 12.47842670244756076349548000211, 13.28954237544223449796890717624, 14.41061957079277895255773494408, 16.02795210342975880387952274685, 16.60440007420358547069389290874, 17.79019340908770328013167042711, 18.36308832667370835894041862456, 19.222316049181623678219722324450, 20.06567942589626869128391897274, 20.56942793067409487516956409388, 22.142544650231954762471517823087, 22.74322285609644169796236955476, 23.496131770511869647671752776572, 24.64952055364090260179568557107

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