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Properties

Label 1-4033-4033.2401-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.739 - 0.672i$
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  + 2-s + (−0.993 + 0.116i)3-s + 4-s + (−0.686 + 0.727i)5-s + (−0.993 + 0.116i)6-s + (0.973 + 0.230i)7-s + 8-s + (0.973 − 0.230i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.993 + 0.116i)12-s + (−0.686 + 0.727i)13-s + (0.973 + 0.230i)14-s + (0.597 − 0.802i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.993 + 0.116i)3-s + 4-s + (−0.686 + 0.727i)5-s + (−0.993 + 0.116i)6-s + (0.973 + 0.230i)7-s + 8-s + (0.973 − 0.230i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.993 + 0.116i)12-s + (−0.686 + 0.727i)13-s + (0.973 + 0.230i)14-s + (0.597 − 0.802i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.739 - 0.672i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.739 - 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.699275398 - 0.6573584614i\)
\(L(\frac12)\) \(\approx\) \(1.699275398 - 0.6573584614i\)
\(L(1)\) \(\approx\) \(1.311624145 + 0.05100402916i\)
\(L(1)\) \(\approx\) \(1.311624145 + 0.05100402916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (-0.686 + 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.686 - 0.727i)T \)
13 \( 1 + (-0.686 + 0.727i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.973 + 0.230i)T \)
31 \( 1 + (-0.0581 - 0.998i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.993 - 0.116i)T \)
53 \( 1 + (0.396 - 0.918i)T \)
59 \( 1 + (0.597 - 0.802i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (-0.993 - 0.116i)T \)
71 \( 1 + T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (0.597 - 0.802i)T \)
83 \( 1 + (-0.835 - 0.549i)T \)
89 \( 1 + (-0.286 - 0.957i)T \)
97 \( 1 + (-0.0581 + 0.998i)T \)
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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

−18.40338999937437118647276892471, −17.67059185096926582180613094644, −17.04737196869311606007343535963, −16.53371865974999497782103389183, −15.72453653887258257903375599437, −15.07075420792119319360261618868, −14.74443585443969515683632263932, −13.53715599041313780786243791474, −12.81940931299856214528095020751, −12.44139494886464712925538628910, −11.93747408645012836697651002984, −11.06883372442830377251430777533, −10.56833730788389498458342947112, −10.0060604643007308088774395041, −8.34454340304812202056617516202, −8.062883443443728263020776742577, −7.14769019798985286378033156594, −6.56984498012647976808536652693, −5.5332199717102531987074604902, −4.967070939194951447248532750312, −4.50645068433938222203903770158, −3.97212347389643408511705395346, −2.63395534968793689445978819452, −1.78218196963735069846297411930, −0.96543921415922910412036060018, 0.44591420381654425592659126135, 1.846752876065522684055479123108, 2.51241285034733205527384440389, 3.500473371128312616494671322572, 4.32918406681666071351383295646, 4.91098714253494033636637525623, 5.44621290215227347253596283852, 6.4486903848954947729775418317, 6.92033516041402351648861930026, 7.64604638702987219901670481802, 8.367619675697918259638995199066, 9.652084660983870524871867604764, 10.58102883279185126150005412917, 11.06689320475056257999790565152, 11.649442488901313753867625914515, 11.880138319519839726366234411616, 12.909235697035850489289879439236, 13.59800130938760725139006080604, 14.39822956290206051479173243064, 15.00687701496754354873461208569, 15.64241019163849128618782206519, 16.11083271749772881984312267899, 16.930028186930209171490344706362, 17.642410774976183701924539608144, 18.38394875620009038885770989656

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