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Properties

Label 1-349-349.16-r0-0-0
Degree $1$
Conductor $349$
Sign $0.784 + 0.619i$
Analytic cond. $1.62074$
Root an. cond. $1.62074$
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  + (0.997 + 0.0721i)2-s + (−0.302 + 0.953i)3-s + (0.989 + 0.143i)4-s + (−0.773 − 0.633i)5-s + (−0.370 + 0.928i)6-s + (0.126 − 0.992i)7-s + (0.976 + 0.214i)8-s + (−0.817 − 0.576i)9-s + (−0.725 − 0.687i)10-s + (0.0541 + 0.998i)11-s + (−0.436 + 0.899i)12-s + (0.750 + 0.661i)13-s + (0.197 − 0.980i)14-s + (0.837 − 0.546i)15-s + (0.958 + 0.284i)16-s + (0.976 − 0.214i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0721i)2-s + (−0.302 + 0.953i)3-s + (0.989 + 0.143i)4-s + (−0.773 − 0.633i)5-s + (−0.370 + 0.928i)6-s + (0.126 − 0.992i)7-s + (0.976 + 0.214i)8-s + (−0.817 − 0.576i)9-s + (−0.725 − 0.687i)10-s + (0.0541 + 0.998i)11-s + (−0.436 + 0.899i)12-s + (0.750 + 0.661i)13-s + (0.197 − 0.980i)14-s + (0.837 − 0.546i)15-s + (0.958 + 0.284i)16-s + (0.976 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.784 + 0.619i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.784 + 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.952585191 + 0.6781229038i\)
\(L(\frac12)\) \(\approx\) \(1.952585191 + 0.6781229038i\)
\(L(1)\) \(\approx\) \(1.619486915 + 0.3708681268i\)
\(L(1)\) \(\approx\) \(1.619486915 + 0.3708681268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.997 + 0.0721i)T \)
3 \( 1 + (-0.302 + 0.953i)T \)
5 \( 1 + (-0.773 - 0.633i)T \)
7 \( 1 + (0.126 - 0.992i)T \)
11 \( 1 + (0.0541 + 0.998i)T \)
13 \( 1 + (0.750 + 0.661i)T \)
17 \( 1 + (0.976 - 0.214i)T \)
19 \( 1 + (0.530 + 0.847i)T \)
23 \( 1 + (0.750 + 0.661i)T \)
29 \( 1 + (-0.436 - 0.899i)T \)
31 \( 1 + (0.267 - 0.963i)T \)
37 \( 1 + (-0.947 + 0.319i)T \)
41 \( 1 + (0.976 + 0.214i)T \)
43 \( 1 + (-0.891 - 0.452i)T \)
47 \( 1 + (0.647 - 0.762i)T \)
53 \( 1 + (-0.994 + 0.108i)T \)
59 \( 1 + (-0.302 + 0.953i)T \)
61 \( 1 + (-0.561 - 0.827i)T \)
67 \( 1 + (0.468 + 0.883i)T \)
71 \( 1 + (-0.302 - 0.953i)T \)
73 \( 1 + (-0.922 + 0.386i)T \)
79 \( 1 + (-0.994 - 0.108i)T \)
83 \( 1 + (0.935 - 0.353i)T \)
89 \( 1 + (-0.0180 - 0.999i)T \)
97 \( 1 + (-0.773 + 0.633i)T \)
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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

−24.529917330475516794968630956722, −23.77247166244461269616855657854, −23.01696852885190959426593813113, −22.31813517794857392264705404441, −21.50405316207669534735582346042, −20.33739902009196578243931777669, −19.23932497913394881727249804872, −18.83038901680189923856808758812, −17.82432757407688793066478990272, −16.38953647370257741596539227224, −15.682484859896490123378222894496, −14.648924512726753318250226778613, −13.92721299168178451829662957919, −12.78869588116720384324749168015, −12.17075700838603659429650961554, −11.21633451205622509247353680391, −10.76709517045242895985225894107, −8.65731568906057751178711631621, −7.769153984709460146239519841220, −6.75438997198207207711769763654, −5.89453590759115338043431464606, −5.07575617452005319733656756702, −3.29987148867235368309385220263, −2.82556718553449135111614191180, −1.222809264126124413742486467501, 1.381258054582377760398414800981, 3.39641466675563598756362590760, 4.05505505742027357830430377433, 4.77365430547370888528919364389, 5.78081149953822416080499112354, 7.13688075294444065908241313286, 7.98638342160176429174830372128, 9.480759543742920913029128422634, 10.45499706230969617301917192383, 11.5121985513586314938115316610, 12.015603104582150272704632617940, 13.22924956100709219716296687621, 14.215102151730967983023145923446, 15.11919883910636994452430143005, 15.888415105349785054227699392626, 16.69500751371906982594424017207, 17.21392919083939173809311872302, 19.02264265380355629341379330392, 20.218454562346422860632053123197, 20.65108075248737365721264502202, 21.218137149604692594133396951391, 22.616584844959157646717534664490, 23.164921155048794813995417930723, 23.57662102429479319982621628179, 24.763072813079325553992146086911

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