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Properties

Label 1-308-308.227-r1-0-0
Degree $1$
Conductor $308$
Sign $0.967 - 0.254i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(33.0991\)
Root analytic conductor: \(33.0991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (1:\ ),\ 0.967 - 0.254i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173736614 - 0.1516264648i\)
\(L(\frac12)\) \(\approx\) \(1.173736614 - 0.1516264648i\)
\(L(1)\) \(\approx\) \(0.8549870094 + 0.1174604991i\)
\(L(1)\) \(\approx\) \(0.8549870094 + 0.1174604991i\)

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

−25.17345605792673819550352600906, −24.03349547021842360105959835936, −23.34460129618561336546796494198, −22.65355785534871104370227256454, −21.725280269013837866805638076407, −20.263740006681162194284427674773, −19.57577719531581970652264568501, −18.80902509388149279780245815327, −17.89987198430238253414420953543, −17.26782405799628082320227789284, −15.790061618656982047103463970175, −15.07919135274187565952850776273, −13.95458745834631097433934090668, −13.10444461081369287044478487142, −12.14287748470081877903779531343, −11.20115163172247238746704933546, −10.48522835819180508998625882994, −8.79335684254796321733295225348, −7.96316160345822720943881719877, −6.9701477870234586869237864638, −6.27603552539651900695238352691, −4.87926468857506172451566248963, −3.3499487562073911999141118727, −2.41999288428137823559611883112, −0.84625981018413152903741303287, 0.49122833555108860228049143078, 2.35622530001170693386109258764, 4.05599470993912638979117003654, 4.340477857534312025101976002301, 5.644675602423639076928591533762, 6.85270027139130908097189710233, 8.47670285137639995740973430016, 8.830673929242865047633279073508, 10.07906335836263365704937622825, 11.10534681572903117871100454240, 11.86921899954141319286600877195, 12.9602063988157353786451667735, 14.164496940813797952838837724284, 15.19945062605584793661122654446, 15.906890312661518079188212536576, 16.72174263223974379082215089023, 17.432515143082127753764499606495, 18.94041572105035009488814846288, 19.71305487228348664313292504993, 20.75535449975102476749277759922, 21.21319889901315458552506918575, 22.3555624850877118284448490426, 23.19124169444774910862148426092, 24.01169482944553224175878730075, 25.00797457112809672491126991553

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