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Properties

Label 1-311-311.13-r0-0-0
Degree $1$
Conductor $311$
Sign $0.945 + 0.324i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (−0.440 − 0.897i)2-s + (−0.440 + 0.897i)3-s + (−0.612 + 0.790i)4-s + (−0.874 − 0.485i)5-s + 6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (−0.612 − 0.790i)9-s + (−0.0506 + 0.998i)10-s + (0.528 − 0.848i)11-s + (−0.440 − 0.897i)12-s + (−0.612 − 0.790i)13-s + (0.347 − 0.937i)14-s + (0.820 − 0.571i)15-s + (−0.250 − 0.968i)16-s + (−0.0506 + 0.998i)17-s + ⋯
L(s)  = 1  + (−0.440 − 0.897i)2-s + (−0.440 + 0.897i)3-s + (−0.612 + 0.790i)4-s + (−0.874 − 0.485i)5-s + 6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (−0.612 − 0.790i)9-s + (−0.0506 + 0.998i)10-s + (0.528 − 0.848i)11-s + (−0.440 − 0.897i)12-s + (−0.612 − 0.790i)13-s + (0.347 − 0.937i)14-s + (0.820 − 0.571i)15-s + (−0.250 − 0.968i)16-s + (−0.0506 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.945 + 0.324i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.945 + 0.324i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6465622980 + 0.1079363993i\)
\(L(\frac12)\) \(\approx\) \(0.6465622980 + 0.1079363993i\)
\(L(1)\) \(\approx\) \(0.6508934136 - 0.04038357461i\)
\(L(1)\) \(\approx\) \(0.6508934136 - 0.04038357461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.440 - 0.897i)T \)
3 \( 1 + (-0.440 + 0.897i)T \)
5 \( 1 + (-0.874 - 0.485i)T \)
7 \( 1 + (0.688 + 0.724i)T \)
11 \( 1 + (0.528 - 0.848i)T \)
13 \( 1 + (-0.612 - 0.790i)T \)
17 \( 1 + (-0.0506 + 0.998i)T \)
19 \( 1 + (-0.874 + 0.485i)T \)
23 \( 1 + (0.979 + 0.201i)T \)
29 \( 1 + (0.347 + 0.937i)T \)
31 \( 1 + (-0.0506 - 0.998i)T \)
37 \( 1 + (0.688 - 0.724i)T \)
41 \( 1 + (0.151 + 0.988i)T \)
43 \( 1 + (0.688 + 0.724i)T \)
47 \( 1 + (0.347 + 0.937i)T \)
53 \( 1 + (0.688 + 0.724i)T \)
59 \( 1 + (0.688 + 0.724i)T \)
61 \( 1 + (-0.874 + 0.485i)T \)
67 \( 1 + (0.151 - 0.988i)T \)
71 \( 1 + (-0.758 - 0.651i)T \)
73 \( 1 + (0.918 + 0.394i)T \)
79 \( 1 + (0.151 + 0.988i)T \)
83 \( 1 + (0.820 + 0.571i)T \)
89 \( 1 + (0.688 - 0.724i)T \)
97 \( 1 + (-0.758 - 0.651i)T \)
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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

−25.031357541254562099758860169152, −24.29060864362532196669426945011, −23.39240526831028768271768702641, −23.09233336832274355872697158709, −22.079582246179319462188089982112, −20.32795142933281589679221617647, −19.41030322524860353537135024566, −18.84779757789648347279248879172, −17.755816542419535004485218912189, −17.20930865189350755002101601471, −16.330201601181913149444609185129, −15.07972923522333395274089052512, −14.37499082776777954831896492684, −13.5127737649512057853156774081, −12.155291293947563634917438912166, −11.29011590484007527102932736406, −10.35805954769481375475998148499, −8.93110013056767444237215093821, −7.86850826390402081468250119654, −6.93172516182684589103263896348, −6.84058891394352463923367547976, −5.0202652350282264625650357871, −4.28439311422965524578732478288, −2.15969355567265517742008743413, −0.688106684211258746566153125551, 1.04758985072310553776917781386, 2.831468511369803927079123530226, 3.92707697545113621248561544128, 4.74112101766735435462022437102, 5.85967152311878797061916097123, 7.824293632834807500798191184308, 8.62447318029160482424808581086, 9.32715045629506521803204525854, 10.72786941279368982276276262626, 11.18551027143652748743500685203, 12.153724819679846707293373705578, 12.811983865560691166998319804537, 14.5738782466582766389892908862, 15.25731472181977236495332938601, 16.52741093112550686507969362676, 17.06253077908202123937081074781, 18.076941465487972198280548472226, 19.24300003937201535164153912002, 19.871965058624032844185607365904, 20.97192475293479455411112556874, 21.4995934442471791050780404048, 22.34348169020024372252158454632, 23.25492004494127500765863091224, 24.31841770411301792420994675930, 25.467512099730890377229864870447

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