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Properties

Label 1-3381-3381.1187-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.745 + 0.666i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.476 + 0.879i)2-s + (−0.546 + 0.837i)4-s + (0.894 + 0.446i)5-s + (−0.996 − 0.0815i)8-s + (0.0339 + 0.999i)10-s + (0.390 + 0.920i)11-s + (0.262 − 0.965i)13-s + (−0.403 − 0.915i)16-s + (0.998 − 0.0543i)17-s + (−0.580 − 0.814i)19-s + (−0.862 + 0.505i)20-s + (−0.623 + 0.781i)22-s + (0.601 + 0.798i)25-s + (0.973 − 0.229i)26-s + (0.452 − 0.891i)29-s + ⋯
L(s)  = 1  + (0.476 + 0.879i)2-s + (−0.546 + 0.837i)4-s + (0.894 + 0.446i)5-s + (−0.996 − 0.0815i)8-s + (0.0339 + 0.999i)10-s + (0.390 + 0.920i)11-s + (0.262 − 0.965i)13-s + (−0.403 − 0.915i)16-s + (0.998 − 0.0543i)17-s + (−0.580 − 0.814i)19-s + (−0.862 + 0.505i)20-s + (−0.623 + 0.781i)22-s + (0.601 + 0.798i)25-s + (0.973 − 0.229i)26-s + (0.452 − 0.891i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.745 + 0.666i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.745 + 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8619427183 + 2.257471356i\)
\(L(\frac12)\) \(\approx\) \(0.8619427183 + 2.257471356i\)
\(L(1)\) \(\approx\) \(1.165845326 + 0.9352243123i\)
\(L(1)\) \(\approx\) \(1.165845326 + 0.9352243123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.476 + 0.879i)T \)
5 \( 1 + (0.894 + 0.446i)T \)
11 \( 1 + (0.390 + 0.920i)T \)
13 \( 1 + (0.262 - 0.965i)T \)
17 \( 1 + (0.998 - 0.0543i)T \)
19 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.452 - 0.891i)T \)
31 \( 1 + (-0.786 + 0.618i)T \)
37 \( 1 + (-0.990 - 0.135i)T \)
41 \( 1 + (0.0611 + 0.998i)T \)
43 \( 1 + (-0.996 + 0.0815i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.534 + 0.844i)T \)
59 \( 1 + (0.0339 + 0.999i)T \)
61 \( 1 + (-0.288 - 0.957i)T \)
67 \( 1 + (0.888 + 0.458i)T \)
71 \( 1 + (-0.101 + 0.994i)T \)
73 \( 1 + (0.938 + 0.346i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (0.742 + 0.670i)T \)
89 \( 1 + (0.440 + 0.897i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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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

−18.64516929487237292717756726214, −18.16448094239454083128915691345, −17.03849348845351241912275139927, −16.68002588306936901079392008200, −15.84837643398420384177267767704, −14.5629160708477554636987464785, −14.37214945337131545974447537347, −13.58454125909106310209854526713, −13.07439378066879420825302215582, −12.13970292576614970594346099440, −11.80506315855428626139351022631, −10.717245107879018131996893137500, −10.31070959404632276933275135448, −9.41781306443850486465327722881, −8.87998531730888324686318020208, −8.23460589512152535945151093107, −6.80456794768302468813915143486, −6.134421028827527030638912257948, −5.47266742567659182918797969770, −4.83276060094151652763401771013, −3.7431261159658903116326196327, −3.348844890545226655639337428305, −2.01536037864852033457267912737, −1.68240523121098766054270178336, −0.637465373958097596500957748777, 1.108292933300744326246917781512, 2.30671391486173234177488508425, 3.05821711636469176153025939455, 3.87429325124394906283423226872, 4.89173161879992205997927646177, 5.4183413492765512040870261301, 6.251406060965447824529482620404, 6.80640681659841908657839679236, 7.54524820708666312093202031570, 8.313265799182044260608774637, 9.18687113432561111310858257849, 9.8313254797403007908807413838, 10.52585398227465677461644326714, 11.50883362807387052679810867695, 12.48438879600730185210002129479, 12.87171865108312485371399869686, 13.71493914610450173077663084709, 14.253837202961427262007665053498, 15.02415649496649536805461994479, 15.36535058674399097893845993645, 16.33936970243453235679571385103, 17.12868048296911618892675787008, 17.5521485429634113244851113085, 18.119666703776263054144382590161, 18.81628121698115073505649707155

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