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Properties

Label 1-349-349.3-r0-0-0
Degree $1$
Conductor $349$
Sign $0.654 + 0.755i$
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  + (0.891 + 0.452i)2-s + (0.935 − 0.353i)3-s + (0.590 + 0.806i)4-s + (−0.968 + 0.250i)5-s + (0.994 + 0.108i)6-s + (0.999 + 0.0361i)7-s + (0.161 + 0.986i)8-s + (0.750 − 0.661i)9-s + (−0.976 − 0.214i)10-s + (−0.907 − 0.419i)11-s + (0.837 + 0.546i)12-s + (0.0180 + 0.999i)13-s + (0.874 + 0.484i)14-s + (−0.817 + 0.576i)15-s + (−0.302 + 0.953i)16-s + (−0.161 + 0.986i)17-s + ⋯
L(s)  = 1  + (0.891 + 0.452i)2-s + (0.935 − 0.353i)3-s + (0.590 + 0.806i)4-s + (−0.968 + 0.250i)5-s + (0.994 + 0.108i)6-s + (0.999 + 0.0361i)7-s + (0.161 + 0.986i)8-s + (0.750 − 0.661i)9-s + (−0.976 − 0.214i)10-s + (−0.907 − 0.419i)11-s + (0.837 + 0.546i)12-s + (0.0180 + 0.999i)13-s + (0.874 + 0.484i)14-s + (−0.817 + 0.576i)15-s + (−0.302 + 0.953i)16-s + (−0.161 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.654 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.474012929 + 1.129649656i\)
\(L(\frac12)\) \(\approx\) \(2.474012929 + 1.129649656i\)
\(L(1)\) \(\approx\) \(2.021164831 + 0.5874437979i\)
\(L(1)\) \(\approx\) \(2.021164831 + 0.5874437979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.891 + 0.452i)T \)
3 \( 1 + (0.935 - 0.353i)T \)
5 \( 1 + (-0.968 + 0.250i)T \)
7 \( 1 + (0.999 + 0.0361i)T \)
11 \( 1 + (-0.907 - 0.419i)T \)
13 \( 1 + (0.0180 + 0.999i)T \)
17 \( 1 + (-0.161 + 0.986i)T \)
19 \( 1 + (0.958 + 0.284i)T \)
23 \( 1 + (-0.0180 - 0.999i)T \)
29 \( 1 + (0.837 - 0.546i)T \)
31 \( 1 + (-0.561 - 0.827i)T \)
37 \( 1 + (-0.856 + 0.515i)T \)
41 \( 1 + (-0.161 - 0.986i)T \)
43 \( 1 + (0.0901 + 0.995i)T \)
47 \( 1 + (-0.796 - 0.605i)T \)
53 \( 1 + (-0.647 - 0.762i)T \)
59 \( 1 + (-0.935 + 0.353i)T \)
61 \( 1 + (-0.0541 + 0.998i)T \)
67 \( 1 + (-0.725 - 0.687i)T \)
71 \( 1 + (-0.935 - 0.353i)T \)
73 \( 1 + (0.530 - 0.847i)T \)
79 \( 1 + (-0.647 + 0.762i)T \)
83 \( 1 + (0.700 + 0.713i)T \)
89 \( 1 + (0.619 - 0.785i)T \)
97 \( 1 + (0.968 + 0.250i)T \)
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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

−24.64059283772658301987798375428, −23.74589769413866822867261138066, −23.03420523191482963550925313081, −21.93123136380793889982319366195, −21.03274083055555724850035798154, −20.19680310644589924178399401301, −20.043991776438021164970641020318, −18.777407615622841184818299076290, −17.85683738116991758936021276192, −15.85111817275294486191906543453, −15.75485944743331736660127875530, −14.7488234007117526123909792064, −13.93477545629736152238690132262, −13.01864986192579573771050132183, −12.06039218552559873464004792006, −11.072168391254522694810363796952, −10.26553050061034133925054129946, −9.03076646933941170134925032977, −7.77651662860761265981803788740, −7.31287699630063329883794567665, −5.094107473589471659138622756581, −4.872649084204637610080017184967, −3.50930265707776245000298655936, −2.781363537017197386198451160781, −1.37334269507768031840070408273, 1.841660558294483977982579847431, 2.97247313958822871435591799430, 3.98472331567196974288237233051, 4.81045442289748295109207704665, 6.30555886432896533920314007336, 7.389575971892913821275227849594, 8.05285958197093623189563939043, 8.69029123842511365399112820036, 10.55203817634329347081101399770, 11.59953944174873954474856706717, 12.33767556005999974360129809312, 13.44460606667128757638500952620, 14.24459004630384359068955249667, 14.924222855344759516746613644314, 15.673294093429990889700610902545, 16.58603821108762557930840771402, 17.97591331325279940450026701344, 18.827163869569229104790766067821, 19.797462497097823753658389271200, 20.796765240719105626348659296909, 21.23978295562077447552905875310, 22.41615976080106434588979890065, 23.569859032212404568073828406702, 24.12870869579266683153805621576, 24.515440318320923179922126819539

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