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Properties

Label 1-308-308.235-r1-0-0
Degree $1$
Conductor $308$
Sign $-0.370 + 0.928i$
Analytic cond. $33.0991$
Root an. cond. $33.0991$
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  + (0.104 − 0.994i)3-s + (0.669 + 0.743i)5-s + (−0.978 − 0.207i)9-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.978 + 0.207i)17-s + (−0.913 + 0.406i)19-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.669 + 0.743i)31-s + (−0.104 − 0.994i)37-s + (0.978 − 0.207i)39-s + (−0.809 − 0.587i)41-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)3-s + (0.669 + 0.743i)5-s + (−0.978 − 0.207i)9-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.978 + 0.207i)17-s + (−0.913 + 0.406i)19-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.669 + 0.743i)31-s + (−0.104 − 0.994i)37-s + (0.978 − 0.207i)39-s + (−0.809 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.370 + 0.928i$
Analytic conductor: \(33.0991\)
Root analytic conductor: \(33.0991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (1:\ ),\ -0.370 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4657373609 + 0.6869999605i\)
\(L(\frac12)\) \(\approx\) \(0.4657373609 + 0.6869999605i\)
\(L(1)\) \(\approx\) \(0.9591710849 + 0.02369872575i\)
\(L(1)\) \(\approx\) \(0.9591710849 + 0.02369872575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−25.00430714197035000513903056398, −23.907292514722915778867183547728, −22.81519442409682331754896059812, −21.921734456155448790546615013551, −21.226088215840179885790162929258, −20.34577201068695141022012560164, −19.78905130738690916817405414310, −18.282977698785520491072445097881, −17.23239647343683871651014626842, −16.735157009205171405480281284923, −15.50784503856595311015147397614, −15.01415626590144686604873672613, −13.56595571512747814829402274672, −13.06882750968162965210449380906, −11.5996382526893926744214680603, −10.67256215651943201537033153469, −9.70174726248292563604848150878, −8.93754021451541906376638637838, −8.05085396464128548351583130811, −6.35324912873632402931025533573, −5.340087294445079141467632986078, −4.541317616995849554807677460425, −3.298960586200064081033969012007, −1.973708845916404613952671455331, −0.219570059655260754216050793394, 1.63863792727899292252098207430, 2.38574570555858946415982616710, 3.742021854996896059536513741897, 5.392626809687274722215867296, 6.61969480799045873889087633675, 6.89504475961482630384310800532, 8.39429760839330795927517716643, 9.21122877482790694730422432171, 10.63078942008446208695290383258, 11.34796082277657515235198430956, 12.61261126764613065759651949279, 13.358217731549886872215106617861, 14.29870736620168087549968379965, 14.91641171974570930950961373931, 16.47307889992556383701820818292, 17.36679334371915693906434574046, 18.25557078887852337514471888545, 18.84329034857396922808823617981, 19.73990916824736830740234418626, 20.87313640790715799889321135255, 21.81676206909679262079918874203, 22.74693389962287739960570314621, 23.59910667068307005075042290592, 24.47633553467502751011640202192, 25.34849315868025065918874352330

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