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Properties

Label 1-308-308.303-r0-0-0
Degree $1$
Conductor $308$
Sign $0.603 + 0.797i$
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  + (0.978 + 0.207i)3-s + (−0.104 + 0.994i)5-s + (0.913 + 0.406i)9-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.104 + 0.994i)31-s + (−0.978 + 0.207i)37-s + (0.913 − 0.406i)39-s + (−0.309 − 0.951i)41-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)3-s + (−0.104 + 0.994i)5-s + (0.913 + 0.406i)9-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.104 + 0.994i)31-s + (−0.978 + 0.207i)37-s + (0.913 − 0.406i)39-s + (−0.309 − 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(1.43034\)
Root analytic conductor: \(1.43034\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (0:\ ),\ 0.603 + 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.575453684 + 0.7836189194i\)
\(L(\frac12)\) \(\approx\) \(1.575453684 + 0.7836189194i\)
\(L(1)\) \(\approx\) \(1.395800563 + 0.3728063402i\)
\(L(1)\) \(\approx\) \(1.395800563 + 0.3728063402i\)

Euler product

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

−24.835314174942893246619387596065, −24.59524419581912722769593596199, −23.582773896181403066919250429416, −22.500955630479702151497874422950, −21.10016044891056734810259673163, −20.70823406635756926515816815712, −19.87847454038325100374593093173, −18.927261927596020208963785698006, −18.11437834915481901376396493994, −16.83178985786549934810240917979, −15.970660183767124567420680013525, −15.14642944933809422503991433025, −13.91639208769363642673115800509, −13.3317259088680435746206972491, −12.38741369723889275283597836058, −11.354740685988309883280373488534, −9.87792017592716030803087916940, −9.00116532310619342539409678396, −8.34162164436535631079353502291, −7.30095496026273274450318303868, −6.06013445651973378019350041969, −4.60297640883627070112755237573, −3.779522100569062815998912255665, −2.34729993690169187598896067536, −1.17582821046235046317544948656, 1.73487654578097460466734120702, 3.03735048111092077343797441540, 3.64968047388089160641930057849, 5.09562682731237743216595803809, 6.59533910585887177206118008714, 7.41998485965720888237946267398, 8.49196672559981067120810619371, 9.40805634349486019703923276601, 10.55997183023219289914067406681, 11.18712533370091255770970669912, 12.73537437296647129094532548007, 13.67254744603825877654450046184, 14.37274320371856639669839876481, 15.53084025886693369820154007082, 15.73916998162091540988705001024, 17.51004459843628593579693452652, 18.27421647049972819204417557508, 19.24606124225150012772110331176, 19.91787336769832380401767795573, 20.895151074671930216408272612936, 21.84077712778062950562066589174, 22.55201095204869100213675460342, 23.68353353521301758384403102559, 24.64219708478890873901879908731, 25.74485605613327985121662101229

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