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Properties

Label 1-308-308.107-r0-0-0
Degree $1$
Conductor $308$
Sign $0.564 + 0.825i$
Analytic cond. $1.43034$
Root an. cond. $1.43034$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(1.43034\)
Root analytic conductor: \(1.43034\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (0:\ ),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266212697 + 0.6678116122i\)
\(L(\frac12)\) \(\approx\) \(1.266212697 + 0.6678116122i\)
\(L(1)\) \(\approx\) \(1.162645913 + 0.3386862917i\)
\(L(1)\) \(\approx\) \(1.162645913 + 0.3386862917i\)

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

−24.975877972336983571600092968025, −24.56666685419444988259903020085, −23.13809640806491073263281388277, −22.74261589933798496508650439622, −21.616556558380237805559320168146, −20.568158862268174979294088505922, −19.59377499908376486018726026418, −18.72537881056173644114335984764, −17.917350257108279630257294548453, −17.3787353776734246222496323559, −16.085319088306333510348117030393, −14.68216700477765285745354520409, −14.19915941217780050196191415254, −13.15101174349812767403609307966, −12.36867688936648687346709028590, −11.23830337814375753404322091327, −10.242631322605531107057194720806, −9.16797983456916691726620362675, −7.87132179446565912417214943528, −7.12940183816055407166845013428, −6.09396868095263654992957470818, −5.2182068146154063917831121911, −3.20948876029132956673662595302, −2.49816452916574350661097667179, −1.06385043898067449029813884061, 1.47155340449677952522874136234, 2.96210173331794707900277403945, 4.19015646855020005437983456459, 5.1746371214727949292113775074, 5.9607965747022914834658154249, 7.554482944890593799799086327753, 8.80619642384732337105469678094, 9.506570275513208536850487034058, 10.24156581200055166381219942491, 11.49153687328409653026603187229, 12.40679112926930281427750300186, 13.73521265174089073985113004569, 14.33919089164475094131030656476, 15.482177752892624268045738790526, 16.526087318669356016019001627816, 16.8931924873969928673112715887, 18.03645497207271747540821630709, 19.30709692239933661133808957688, 20.24189523879763734087370331607, 21.11312926568467099071025299288, 21.56860394606122251444579691192, 22.56582746588324532567039637241, 23.63911138831322277018367957522, 24.604778292256102365561140887600, 25.5977131685169271877933139404

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