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Properties

Label 1-349-349.121-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.287 + 0.957i$
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  + (0.725 + 0.687i)2-s + (0.647 + 0.762i)3-s + (0.0541 + 0.998i)4-s + (0.796 − 0.605i)5-s + (−0.0541 + 0.998i)6-s + (0.856 + 0.515i)7-s + (−0.647 + 0.762i)8-s + (−0.161 + 0.986i)9-s + (0.994 + 0.108i)10-s + (−0.976 − 0.214i)11-s + (−0.725 + 0.687i)12-s + (−0.267 − 0.963i)13-s + (0.267 + 0.963i)14-s + (0.976 + 0.214i)15-s + (−0.994 + 0.108i)16-s + (0.647 + 0.762i)17-s + ⋯
L(s)  = 1  + (0.725 + 0.687i)2-s + (0.647 + 0.762i)3-s + (0.0541 + 0.998i)4-s + (0.796 − 0.605i)5-s + (−0.0541 + 0.998i)6-s + (0.856 + 0.515i)7-s + (−0.647 + 0.762i)8-s + (−0.161 + 0.986i)9-s + (0.994 + 0.108i)10-s + (−0.976 − 0.214i)11-s + (−0.725 + 0.687i)12-s + (−0.267 − 0.963i)13-s + (0.267 + 0.963i)14-s + (0.976 + 0.214i)15-s + (−0.994 + 0.108i)16-s + (0.647 + 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564006828 + 2.101636330i\)
\(L(\frac12)\) \(\approx\) \(1.564006828 + 2.101636330i\)
\(L(1)\) \(\approx\) \(1.615547724 + 1.220937072i\)
\(L(1)\) \(\approx\) \(1.615547724 + 1.220937072i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.725 + 0.687i)T \)
3 \( 1 + (0.647 + 0.762i)T \)
5 \( 1 + (0.796 - 0.605i)T \)
7 \( 1 + (0.856 + 0.515i)T \)
11 \( 1 + (-0.976 - 0.214i)T \)
13 \( 1 + (-0.267 - 0.963i)T \)
17 \( 1 + (0.647 + 0.762i)T \)
19 \( 1 + (-0.370 - 0.928i)T \)
23 \( 1 + (0.267 + 0.963i)T \)
29 \( 1 + (-0.725 - 0.687i)T \)
31 \( 1 + (0.468 - 0.883i)T \)
37 \( 1 + (0.267 + 0.963i)T \)
41 \( 1 + (0.647 - 0.762i)T \)
43 \( 1 + (-0.976 - 0.214i)T \)
47 \( 1 + (0.947 + 0.319i)T \)
53 \( 1 + (-0.907 - 0.419i)T \)
59 \( 1 + (-0.647 - 0.762i)T \)
61 \( 1 + (0.725 - 0.687i)T \)
67 \( 1 + (-0.370 + 0.928i)T \)
71 \( 1 + (-0.647 + 0.762i)T \)
73 \( 1 + (-0.856 - 0.515i)T \)
79 \( 1 + (-0.907 + 0.419i)T \)
83 \( 1 + (0.796 - 0.605i)T \)
89 \( 1 + (0.561 - 0.827i)T \)
97 \( 1 + (-0.796 - 0.605i)T \)
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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

−24.54403841079863174442996609160, −23.49199845086547377546394440459, −23.06623521724449582404343635491, −21.67882575841746235498371390364, −20.91172326824694949842635118860, −20.50495089716202774219292937390, −19.24706961801806891418919949756, −18.4410758873970606462234930433, −17.99164582445362329978942004159, −16.559864154721339064986012234895, −14.93307686921106875562820773180, −14.36531042583452750199738096989, −13.83689291735175107886178442137, −12.907133083801357969355274173660, −12.00216867611242845345026479121, −10.88710448890872352821752583741, −10.07851864444661683120735156755, −9.01698802293148040354385142740, −7.60707711037568339887413523842, −6.76091804482281932060564489680, −5.62716653930924547591972978743, −4.48428772752614052848666780566, −3.12795444927362127841276122582, −2.22542270919494985085778892395, −1.39177115105895265333641487364, 2.12475806913964935368523444042, 3.04570608788341642644344480921, 4.446527289712222996226548146613, 5.27951347539647073848605997028, 5.81132971079283329363787373873, 7.71850750815655166899374490593, 8.2571530431908286593028551050, 9.21054978381501282838349520165, 10.34756506000896757984971201114, 11.50328648269282114344608378587, 12.90517813760754294471112827744, 13.38884849412902451628446580031, 14.44043418050740491540574855211, 15.252656253769111971149293924243, 15.776744190061763130990415102997, 17.09930791509751599893291421275, 17.48972283615763359566604282268, 18.85317219973074267421690219017, 20.39894186955044587777282604779, 20.85597863903350126159190826380, 21.63331393542927132248192606324, 22.13193356085849787019828420577, 23.553579689035130468155701069120, 24.321686087745965129317085036, 25.14759489147302604116373209184

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