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Properties

Label 1-308-308.171-r1-0-0
Degree $1$
Conductor $308$
Sign $0.894 + 0.446i$
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  + (0.669 − 0.743i)3-s + (−0.913 + 0.406i)5-s + (−0.104 − 0.994i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.913 + 0.406i)31-s + (0.669 + 0.743i)37-s + (−0.104 + 0.994i)39-s + (0.309 + 0.951i)41-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)3-s + (−0.913 + 0.406i)5-s + (−0.104 − 0.994i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.913 + 0.406i)31-s + (0.669 + 0.743i)37-s + (−0.104 + 0.994i)39-s + (0.309 + 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(33.0991\)
Root analytic conductor: \(33.0991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (1:\ ),\ 0.894 + 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.722453987 + 0.4059323357i\)
\(L(\frac12)\) \(\approx\) \(1.722453987 + 0.4059323357i\)
\(L(1)\) \(\approx\) \(1.133035634 - 0.06102574804i\)
\(L(1)\) \(\approx\) \(1.133035634 - 0.06102574804i\)

Euler product

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

−24.81930608568367382014306877490, −24.41229263640777155246108490959, −22.952735440383780831718794447688, −22.42747170802768539991570455098, −21.20984268046987806766016619314, −20.44768812457594451154300059568, −19.713249504370181909206187881653, −19.02901728237778060567342119117, −17.64891615119337168243666541977, −16.55995776725194380509571978007, −15.71249690164509316485054292647, −15.16384807949131766884766561777, −14.07481028498457978760840853553, −13.1081114066457615960056649321, −11.896343822161337969046030431930, −11.07987075432339649639172234201, −9.785136862325097406503142896817, −9.130484318556461853119269874969, −7.89550923444857169060518606676, −7.362994829967610181534659274292, −5.40914813351480300199616584179, −4.59342163232149832795014887225, −3.517278981260048413198492715924, −2.52452896864604391013178824086, −0.58071792469182678679638194923, 1.05206054279332149584390884678, 2.50439651568663088588368265072, 3.46848213835086067087286924877, 4.63358952231035534278276287415, 6.33571567055637516369890956631, 7.1863625508848235670876797144, 8.00462918482068757647808258443, 8.91711739291618172361350886768, 10.13396960570042272235253594924, 11.4175549616019287311984307503, 12.22002934214093227529737208680, 13.06440551430376815775630721545, 14.36515121475364095002086054354, 14.77316527275337252079603998714, 15.87956089020173182083546326780, 17.00733632297864268645672179342, 18.17813372682049585089798854923, 18.91435132168969488929474060228, 19.64999211686038763244312674709, 20.319293980367306768640223837408, 21.52401716607278729244347443114, 22.59113895495852712615004306534, 23.50718413056533506942180193274, 24.24726480053800561490799059286, 24.951036837877926497128557965787

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