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Properties

Label 1-4033-4033.2879-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.624 - 0.781i$
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  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + 10-s + 11-s + (−0.939 − 0.342i)12-s + (0.939 − 0.342i)13-s − 14-s + (−0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + 10-s + 11-s + (−0.939 − 0.342i)12-s + (0.939 − 0.342i)13-s − 14-s + (−0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.624 - 0.781i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.624 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4773507350 - 0.9927994740i\)
\(L(\frac12)\) \(\approx\) \(0.4773507350 - 0.9927994740i\)
\(L(1)\) \(\approx\) \(0.7445004049 - 0.2741577809i\)
\(L(1)\) \(\approx\) \(0.7445004049 - 0.2741577809i\)

Euler product

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

−18.971990975118260605520679846934, −17.86565008472197108415542065958, −17.49939569958126094400392865795, −16.587063096431109264692374482973, −16.26976573763501072451169565775, −15.31984085786739244255433530528, −14.70903893536801570345263844282, −14.12162980198292067864331851064, −13.24892966767425191133561124102, −12.17922427755164930027520146741, −11.48059721424285673199043725070, −11.03401041668178434876278853076, −10.5583245014786432500037573193, −9.89519051469078252441135737381, −8.954159734114284966333057239239, −8.34144474078421561047897369478, −7.970430124070841815770543984715, −6.847562197641127597522149259430, −6.32192224066169411018836224491, −4.85097171417380473321645435150, −4.12383103421319414696998137086, −3.658144144593606235820112235475, −3.122897189198507654090681372785, −1.84811168122400052471852230107, −1.168447454407964820049033336485, 0.477905016386574164592294262153, 1.088159088200171894553399863325, 1.97007353398004452687576411159, 2.786707443994285671877783047336, 4.11249785926530484609889936164, 4.89353778811599126264263468654, 5.712239545393387521326603683308, 6.48150781815826596249007318466, 7.06489404042822898323780414033, 7.90889201494585258228260763772, 8.52809175247036679854000928834, 8.759213492717983673210725450393, 9.517478674274710171946125489657, 10.83267785871474490115794937606, 11.482088268715036572611611826217, 11.83195684147323672739979277420, 12.7318189343765607005722631017, 13.60308624826789856432523361816, 14.17150715691051897650321673631, 15.17140798698123086812551937993, 15.30166151612439859632956408277, 16.31490186387321742247100669601, 17.07268587494515081950548780560, 17.4702245376802238107929670135, 18.32990500740989451947753761145

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