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Properties

Label 1-344-344.197-r0-0-0
Degree $1$
Conductor $344$
Sign $0.570 + 0.821i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.570 + 0.821i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6942662346 + 0.3631803036i\)
\(L(\frac12)\) \(\approx\) \(0.6942662346 + 0.3631803036i\)
\(L(1)\) \(\approx\) \(0.8039035910 + 0.02134263666i\)
\(L(1)\) \(\approx\) \(0.8039035910 + 0.02134263666i\)

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

−24.65432270889654768299656853641, −24.010458894330451139444231057200, −22.76413487092079650906274655080, −22.02213815115895886319360348217, −21.336177459322713356924602017160, −20.47922764540182414622141972042, −19.54536333347804412108148343622, −18.58128514436040958932818296075, −17.29886556351190761765693060935, −16.53131408609308743504875486853, −15.96869584148869638816647063043, −15.09270737415301727188357613243, −13.94500240907165086466138963274, −12.81333853774114761651264366148, −11.77658818865399490359801644266, −11.274145524005001399271910926206, −9.55782746267002869572431447962, −9.35508986518102141824104636216, −8.339917916104972110262149180786, −6.75454666906110455990422407565, −5.48592219527719292298332586823, −5.00007018508140564889258490252, −3.7317200223362880235818607470, −2.56859480724959624303816137504, −0.536958925674514513945648677217, 1.38859737600068897070866224221, 2.6872415774072080562946624443, 3.80615250023592496544255228710, 5.34062337874690313368425125068, 6.47338269427509297233890011997, 7.28947220103274891190095960586, 7.75462053006674503304128278735, 9.535699425655707183165615120965, 10.48000816542827368680105087961, 11.244994591194422247160477107260, 12.50695381524050546155482351067, 13.00200624217141766487271559670, 14.31857123710128463512780703302, 14.790324235164830633498657027774, 16.23062768896717789919915160934, 17.276578753670703508048660287766, 17.80796151620167300878278845467, 18.826680287009049557137642393712, 19.62927767412577718381675025646, 20.24249013663135060223798423961, 21.82780796703640564596630580156, 22.674846412353785917125797137936, 23.11622961867667728129534744451, 23.98279248546264811378816106055, 25.15547537531681304327171088569

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