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Properties

Label 1-4033-4033.1135-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.112 - 0.993i$
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  + (−0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.939 + 0.342i)5-s + (0.5 + 0.866i)6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)12-s + (−0.173 − 0.984i)13-s + (0.5 − 0.866i)14-s + (0.939 − 0.342i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.939 + 0.342i)5-s + (0.5 + 0.866i)6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)12-s + (−0.173 − 0.984i)13-s + (0.5 − 0.866i)14-s + (0.939 − 0.342i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.112 - 0.993i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.112 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019483651 - 1.140849767i\)
\(L(\frac12)\) \(\approx\) \(1.019483651 - 1.140849767i\)
\(L(1)\) \(\approx\) \(1.131289304 - 0.06869805136i\)
\(L(1)\) \(\approx\) \(1.131289304 - 0.06869805136i\)

Euler product

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

−18.65554891453647810688929125326, −18.4144065638911939547657069727, −17.14232958658740139961062227492, −16.71022531988923993551072298243, −16.162948883452588215074255177257, −14.94172512706721862188554103736, −14.56998248207252033579607764237, −13.63722844199798348324864536016, −13.26465758937127181960889364722, −12.37654723034503717132185754862, −12.09301934121566586329348807801, −10.65643037334333331879691433989, −10.23872234558227428712907921966, −9.73867989136003496360303458410, −9.150051727293548734236878900655, −8.619512195251138975863415240089, −7.75688761945371798138708964349, −6.75831248509548936204486719872, −5.67096101422955355716919302130, −4.99154649681807970460109469844, −4.25673946823048514142392928144, −3.44936741320524115692858272658, −2.70405348629112525386675424980, −2.05154968353763245214312963571, −1.39707933480204192428170929660, 0.4117486837914256146699455624, 1.20312470412519266977899221137, 2.47742505300603387615159708618, 3.18442946246239344160736062012, 3.76590186942480224140835774593, 5.28970747264384967876455618936, 5.63084087872391679668479760119, 6.51101294931585512764745472991, 7.05681626728737348279514563533, 7.68977329299652620338043187051, 8.45074932540867260434890763168, 9.20036841100703172484345146513, 9.82055006906804987685268762491, 10.26266360003522094884438030864, 11.34371119297499033756282025060, 12.6461722945657451549433815359, 13.250581111990058569667588699880, 13.449695290452566523640660759916, 14.117136127491564739777991562777, 14.965415595383506980293987625296, 15.41445152711024682054247516197, 16.245132074681842167686617667085, 17.09838203946773347481447190241, 17.48160434207731648655171733463, 18.347657569272890038664585282280

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