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Properties

Label 1-4033-4033.147-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.00883 + 0.999i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (0.5 + 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.766 + 0.642i)5-s + (−0.173 − 0.984i)6-s + (0.766 − 0.642i)7-s − 8-s + (0.766 + 0.642i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)12-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.939 − 0.342i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.766 + 0.642i)5-s + (−0.173 − 0.984i)6-s + (0.766 − 0.642i)7-s − 8-s + (0.766 + 0.642i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)12-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.939 − 0.342i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00883 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00883 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.00883 + 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.00883 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8130261971 + 0.8202453766i\)
\(L(\frac12)\) \(\approx\) \(0.8130261971 + 0.8202453766i\)
\(L(1)\) \(\approx\) \(0.7391324303 + 0.4439580800i\)
\(L(1)\) \(\approx\) \(0.7391324303 + 0.4439580800i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
41 \( 1 + T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−18.168054398048207446892681582816, −17.86200171793069754193528245734, −17.05001934974713983883388420275, −15.960458128839790996997592373616, −15.630893032729063421273606412463, −15.0461787600623773099342363859, −14.15545907290558730270392079666, −13.23093529444199031292869550171, −12.57519511886042182718076318113, −12.02993047919760987143672656945, −11.52503115562719348061051382881, −10.94437297826985990840661084787, −10.27751528950877130743710349512, −9.364176362395837885675319482995, −8.85394354868542630426094901870, −7.76196757653103205049022659564, −7.1571083362644287634184567224, −5.69439889108385831228489314651, −5.35949755807793940876273990491, −4.879105342790152130045174229511, −4.22208437192912457404505221030, −3.14502624999743450876180664540, −2.54317746903484770276221899513, −1.20886140233361522869476349387, −0.611270786251534332392208224029, 0.592335137238428794939845183610, 1.9286217638158701474800733048, 2.928821843835948506190797904610, 4.08645165429630840499178660320, 4.42581799970012095325967081280, 5.26838705279564134433583188289, 5.909936265228557773391559603797, 6.98689641837952651046260352002, 7.165395904086468639264701029913, 7.935551057307851680859582856226, 8.38888203370544865692482116455, 9.88969254173377192673552325634, 10.443040620519793040128769000745, 11.269965175359032941083524270328, 11.83962939823977940393839530305, 12.58431089293915158055167696579, 13.028037035098837261459633430447, 14.17496296327754470275081794971, 14.52171911158249419183953319238, 15.17369760535545058658435763283, 16.17147820126268376779259700297, 16.404718646435356935065299803092, 17.2817681066820138305398178107, 17.75670882039282345766924072169, 18.542543785948841744858143251915

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