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Properties

Label 1-4033-4033.152-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.998 + 0.0461i$
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 + (0.766 − 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.766 + 0.642i)5-s + (0.173 − 0.984i)6-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (−0.173 − 0.984i)11-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.766 + 0.642i)5-s + (0.173 − 0.984i)6-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (−0.173 − 0.984i)11-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.998 + 0.0461i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.998 + 0.0461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06861595768 - 2.973006595i\)
\(L(\frac12)\) \(\approx\) \(-0.06861595768 - 2.973006595i\)
\(L(1)\) \(\approx\) \(1.275466880 - 1.324749170i\)
\(L(1)\) \(\approx\) \(1.275466880 - 1.324749170i\)

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 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - T \)
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

−18.590809163800937557930542031305, −18.24521960989486015894490304571, −17.0623808727483613046683098403, −16.64852035677016563768957058456, −15.73338565990082540548432273096, −15.4004079522081548384518926730, −14.82528442637432312487292566346, −14.36507328821435123078022840254, −13.35889865692610746390588870410, −12.765745863013260807942935896825, −12.16835461383643581977221124482, −11.39597003981718723450165730688, −10.68621579994979159515208331210, −9.5217355610076641922764543392, −8.934217592259106734306978393810, −8.247180395420003434205155004809, −7.56145172976096918301161194869, −7.33592354128793571847895168989, −5.76535995734967943397819913382, −5.15422252343030823939584374049, −4.73026134968331699758072420220, −3.94662157560837725429146737904, −3.20207082622554713489041853158, −2.47676577287808257411086345144, −1.45232039306948208174246760167, 0.54096046885994223202601191688, 1.446442385900910627488685435816, 2.229247641752343449010692351518, 3.06538909360137171432204132811, 3.78068052868188952599504444921, 4.16458316568141038028882269572, 5.25462048411315960623403739994, 6.20014029097173844247385517846, 7.05943376207447001165709264576, 7.33884571341530373945903826895, 8.50380566464208709803513443506, 8.89183368404036159661995319389, 10.02924975112980936540278204491, 10.92859436852844734886393298789, 11.229979350684380826591690019762, 11.90000159612804582775737578377, 12.75532817940195905857625615963, 13.45913503468797441961811664777, 14.004798787132360277422965226, 14.49666643342132600244184522722, 15.14173981522909280371563112265, 15.713873061414319923731092599347, 16.748537303350504460225838669211, 17.65683759561698671281470494862, 18.556073387414212233190963085943

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