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Properties

Label 1-304-304.67-r0-0-0
Degree $1$
Conductor $304$
Sign $0.692 - 0.721i$
Analytic cond. $1.41177$
Root an. cond. $1.41177$
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  + (0.984 + 0.173i)3-s + (0.642 − 0.766i)5-s + (−0.5 − 0.866i)7-s + (0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (0.984 − 0.173i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (−0.342 − 0.939i)21-s + (0.766 − 0.642i)23-s + (−0.173 − 0.984i)25-s + (0.866 + 0.5i)27-s + (−0.342 + 0.939i)29-s + (−0.5 − 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)3-s + (0.642 − 0.766i)5-s + (−0.5 − 0.866i)7-s + (0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (0.984 − 0.173i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (−0.342 − 0.939i)21-s + (0.766 − 0.642i)23-s + (−0.173 − 0.984i)25-s + (0.866 + 0.5i)27-s + (−0.342 + 0.939i)29-s + (−0.5 − 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.692 - 0.721i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.692 - 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.665942563 - 0.7101140658i\)
\(L(\frac12)\) \(\approx\) \(1.665942563 - 0.7101140658i\)
\(L(1)\) \(\approx\) \(1.455251384 - 0.3030374983i\)
\(L(1)\) \(\approx\) \(1.455251384 - 0.3030374983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.984 + 0.173i)T \)
5 \( 1 + (0.642 - 0.766i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.342 + 0.939i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−25.27117012605425923699670150414, −25.08016508329509759145518767814, −23.65436004348195691562572639981, −22.717873087562999732292445085318, −21.63035245330188210554264775728, −21.07123511585388315758396854685, −20.03306981094624552504878369300, −18.93576590685486426691883759992, −18.40411406746446964242026488219, −17.63975368655502953592569039756, −15.90422977910187189170130908487, −15.37500962461973798135499419779, −14.45201345150319819115449015519, −13.34313410266638515136476701684, −12.95841611709111726181186334977, −11.45116981863038874769280538919, −10.301723286284484481465327921388, −9.38073215113362297181635303475, −8.63004472857910060530152765547, −7.334358933052589332999377128068, −6.48587699035500067507013302455, −5.31323818053432223759089837924, −3.682517555922208788895853669318, −2.66519109652193950042132103108, −1.91830575252523735422287660494, 1.15348044054989975030891533747, 2.528645007151535029662210969028, 3.69571549010758212143974610858, 4.72024021823477167102221897730, 6.039727220948316446409194313042, 7.26340187294027255357875887518, 8.449063761093098075319597523722, 9.04406732366989713307787596962, 10.20048129710271429024250413116, 10.89047641581364129107212780517, 12.8216737099246121482129272212, 13.267574453371218499326318268294, 13.899304845008223075680655334543, 15.18946015116584872113420768002, 16.1277263326332884316043518184, 16.81012191378266145611731059798, 18.07848089750745699496024933101, 18.97718105222184432388045727411, 20.12836919842209123954232793811, 20.56433206894956538185687712772, 21.35675750558869695735651156366, 22.3896980478559151288834941199, 23.72317832270208473221137814389, 24.29985540812806957297910532326, 25.42451836192978691211025515459

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