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Properties

Label 1-344-344.141-r1-0-0
Degree $1$
Conductor $344$
Sign $-0.958 - 0.284i$
Analytic cond. $36.9679$
Root an. cond. $36.9679$
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  + (0.365 − 0.930i)3-s + (0.0747 + 0.997i)5-s + (0.5 − 0.866i)7-s + (−0.733 − 0.680i)9-s + (0.222 − 0.974i)11-s + (−0.826 + 0.563i)13-s + (0.955 + 0.294i)15-s + (0.0747 − 0.997i)17-s + (−0.733 + 0.680i)19-s + (−0.623 − 0.781i)21-s + (0.955 − 0.294i)23-s + (−0.988 + 0.149i)25-s + (−0.900 + 0.433i)27-s + (0.365 + 0.930i)29-s + (−0.988 − 0.149i)31-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)3-s + (0.0747 + 0.997i)5-s + (0.5 − 0.866i)7-s + (−0.733 − 0.680i)9-s + (0.222 − 0.974i)11-s + (−0.826 + 0.563i)13-s + (0.955 + 0.294i)15-s + (0.0747 − 0.997i)17-s + (−0.733 + 0.680i)19-s + (−0.623 − 0.781i)21-s + (0.955 − 0.294i)23-s + (−0.988 + 0.149i)25-s + (−0.900 + 0.433i)27-s + (0.365 + 0.930i)29-s + (−0.988 − 0.149i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.958 - 0.284i$
Analytic conductor: \(36.9679\)
Root analytic conductor: \(36.9679\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (1:\ ),\ -0.958 - 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1749026839 - 1.204059360i\)
\(L(\frac12)\) \(\approx\) \(0.1749026839 - 1.204059360i\)
\(L(1)\) \(\approx\) \(0.9619604996 - 0.4318297037i\)
\(L(1)\) \(\approx\) \(0.9619604996 - 0.4318297037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (0.365 - 0.930i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (-0.733 + 0.680i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (-0.988 - 0.149i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.826 - 0.563i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (0.733 - 0.680i)T \)
71 \( 1 + (-0.955 - 0.294i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.365 + 0.930i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
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

−25.18229948070887965332957830940, −24.37945631086660969196450105765, −23.289399207765449534708041927438, −22.14651757009736446397073046945, −21.46420070776955177064648848648, −20.746517186192671053889105398735, −19.90013169166065065695634866962, −19.16148741040177020449282151264, −17.42372615192562691193872950475, −17.26519014341729968057548791632, −15.93286883445947677710188932588, −15.09826414827203340804740883420, −14.66158758729045950490676894649, −13.15103220354688817107018420069, −12.39749890282917968860048017417, −11.351420117432517758079277519971, −10.16806650450161951199353714350, −9.28918807887121203355855413586, −8.59476151232758876829577982318, −7.65791951332568706852262191529, −5.90939199116072842577846686508, −4.90188197593492488498568153553, −4.37508817471688460281685211563, −2.80204013849474226236455476591, −1.6905582928789318988278547473, 0.30884205363122936432218469113, 1.71415633694361912365293134866, 2.843786275672897193724772477319, 3.86470066258734877816616455852, 5.45921117912332346042614752140, 6.806344939550719378809241380982, 7.17430972363507349441320576197, 8.26334278400481708199276009849, 9.37492180271198645826599739690, 10.70967700214569654176680407884, 11.36629375461051565724173934121, 12.45739610151630750244198550832, 13.63667703352520601227893539375, 14.2993192837535835793763907790, 14.745490755338932515501157416998, 16.39173809097357771113198256516, 17.26510700247631891484376404502, 18.17300707379803765157246851183, 18.97202564331886552149653619179, 19.579273318706526566056894874154, 20.666007610013038476845450625359, 21.59919004403745220412193602753, 22.71902888467765369718780456258, 23.450494546760016559921488445461, 24.27341341048049505769792437203

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