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Properties

Label 1-1476-1476.139-r1-0-0
Degree $1$
Conductor $1476$
Sign $-0.690 - 0.723i$
Analytic cond. $158.618$
Root an. cond. $158.618$
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)5-s + (0.104 + 0.994i)7-s + (−0.669 − 0.743i)11-s + (−0.104 + 0.994i)13-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + (0.104 − 0.994i)23-s + (−0.104 − 0.994i)25-s + (−0.978 − 0.207i)29-s + (0.978 − 0.207i)31-s + (0.809 + 0.587i)35-s + (0.309 + 0.951i)37-s + (−0.913 + 0.406i)43-s + (−0.913 + 0.406i)47-s + (−0.978 + 0.207i)49-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)5-s + (0.104 + 0.994i)7-s + (−0.669 − 0.743i)11-s + (−0.104 + 0.994i)13-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + (0.104 − 0.994i)23-s + (−0.104 − 0.994i)25-s + (−0.978 − 0.207i)29-s + (0.978 − 0.207i)31-s + (0.809 + 0.587i)35-s + (0.309 + 0.951i)37-s + (−0.913 + 0.406i)43-s + (−0.913 + 0.406i)47-s + (−0.978 + 0.207i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $-0.690 - 0.723i$
Analytic conductor: \(158.618\)
Root analytic conductor: \(158.618\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1476,\ (1:\ ),\ -0.690 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5250322682 - 1.226426065i\)
\(L(\frac12)\) \(\approx\) \(0.5250322682 - 1.226426065i\)
\(L(1)\) \(\approx\) \(1.063395627 - 0.1903474420i\)
\(L(1)\) \(\approx\) \(1.063395627 - 0.1903474420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (0.669 - 0.743i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.913 + 0.406i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.669 + 0.743i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−20.91254823472528973728983018427, −19.96764085681783465624710442111, −19.38505062501268210141088968605, −18.20151518225434560913339830253, −17.887279252992859091798111341985, −17.13126664199767905588176105697, −16.36950873893958507150304641775, −15.18530380734374450573024908639, −14.86297832788296949611283897133, −13.822317607354718217419727395405, −13.301007058302418051718950694048, −12.53609008269845802144447868673, −11.43925878446804525727009270090, −10.50321728101007728761578848113, −10.18442900362199170088746968495, −9.464719102401126371718553565040, −8.03634603393137500845106016014, −7.54127762974617811385791282651, −6.76341877636147173467779191184, −5.71380151386394925728101715192, −5.11507222733531382955362233698, −3.81728712693585512601049669106, −3.17509629381053397498754236564, −2.0456885699619291735553370803, −1.17041457976141098191652674077, 0.24600260290661617585440009760, 1.35915341126331025159599274865, 2.40206588083568434218689814298, 3.06110099511099275938582633108, 4.591403340712908934291772090224, 5.11400091770492187071553354485, 5.926459062032291772983905087627, 6.71191071003107870532108363652, 7.94331219300150298613760940279, 8.63065096337667191496882672098, 9.39224834282311244147707459008, 9.90396321337088895328768921973, 11.21493643657469531742900608041, 11.76234813693908898812002060389, 12.5932257570867546884362727073, 13.44553204228469901328580382067, 13.96313335269480644543117976236, 14.91116229764486422276817549381, 15.881775109934121104806663625852, 16.376116331436614516232013737081, 17.10541485274462207456062966346, 18.15629284335774216257418352339, 18.5596807433681392101591133979, 19.32019039375768396299558245249, 20.55613379337443692456958719634

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