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Properties

Label 1-351-351.236-r0-0-0
Degree $1$
Conductor $351$
Sign $0.821 - 0.570i$
Analytic cond. $1.63003$
Root an. cond. $1.63003$
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  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.342 − 0.939i)5-s + (0.642 + 0.766i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)10-s + (−0.642 − 0.766i)11-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)16-s + 17-s + (0.866 + 0.5i)19-s + (−0.642 − 0.766i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (−0.766 + 0.642i)25-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.342 − 0.939i)5-s + (0.642 + 0.766i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)10-s + (−0.642 − 0.766i)11-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)16-s + 17-s + (0.866 + 0.5i)19-s + (−0.642 − 0.766i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (−0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.821 - 0.570i$
Analytic conductor: \(1.63003\)
Root analytic conductor: \(1.63003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 351,\ (0:\ ),\ 0.821 - 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7767839103 - 0.2435111821i\)
\(L(\frac12)\) \(\approx\) \(0.7767839103 - 0.2435111821i\)
\(L(1)\) \(\approx\) \(0.7290489462 - 0.07803628349i\)
\(L(1)\) \(\approx\) \(0.7290489462 - 0.07803628349i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.984 - 0.173i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 - T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + iT \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.642 + 0.766i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.984 - 0.173i)T \)
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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

−25.26256729770427871623849682731, −23.96556247950414531898087610336, −23.326514980726817804959160446301, −22.19432253344257571048882912232, −21.11136724443721711821674439586, −20.32631162159748982801021584921, −19.60465841137912982190143549767, −18.3573357535053526786598179237, −18.154045982960719123369375673999, −17.01036344568614725281653022826, −16.14591340770859600787432330125, −15.040336870774951882036212963138, −14.41994052716885545021414492318, −12.984518749729335550563247979603, −11.83100135160104313397894450057, −10.95749688449080863067497443390, −10.353840013444703202483136267967, −9.40165180835017340856366234476, −7.9489052051898159495449029822, −7.463756405878638230191558095250, −6.61838903833290012193793875834, −5.03388745635366755956900670002, −3.544325941131012800140854785658, −2.562889049426688428166507471198, −1.17068787617770532876621317251, 0.84420711533204388917887332981, 2.06852920565354515766585676352, 3.47851691225645944093654950161, 5.31829631152069374349347984282, 5.68813322236855743423534409953, 7.47586887025688667242556782396, 8.09580502540079512340698657663, 8.93370121294486617348234523633, 9.77998947693749810704926229538, 11.09129662213973761297264536538, 11.778609778793557721583531823671, 12.69621888761076882088024690007, 14.07503161197535360955498857341, 15.22442038249494978710202526035, 15.93517618445428110668328190349, 16.72589074985459094804847332378, 17.58025342113790622389324818695, 18.683101152159554821049365999553, 19.0914074414679995797099913457, 20.43698925809840008118689448687, 20.86761826081727357364023636831, 21.794853079900888953883778326029, 23.456855001561741644645083407930, 24.001397694920329518318271968417, 24.927263013862672015859208713671

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