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Properties

Label 1-4033-4033.1683-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.265 - 0.964i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + i·2-s + (0.835 + 0.549i)3-s − 4-s + (0.802 − 0.597i)5-s + (−0.549 + 0.835i)6-s + (0.396 − 0.918i)7-s − i·8-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)10-s + (−0.597 + 0.802i)11-s + (−0.835 − 0.549i)12-s + (0.802 − 0.597i)13-s + (0.918 + 0.396i)14-s + (0.998 − 0.0581i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + i·2-s + (0.835 + 0.549i)3-s − 4-s + (0.802 − 0.597i)5-s + (−0.549 + 0.835i)6-s + (0.396 − 0.918i)7-s − i·8-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)10-s + (−0.597 + 0.802i)11-s + (−0.835 − 0.549i)12-s + (0.802 − 0.597i)13-s + (0.918 + 0.396i)14-s + (0.998 − 0.0581i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.265 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.082825219 - 0.8252327795i\)
\(L(\frac12)\) \(\approx\) \(1.082825219 - 0.8252327795i\)
\(L(1)\) \(\approx\) \(1.230213964 + 0.5147360576i\)
\(L(1)\) \(\approx\) \(1.230213964 + 0.5147360576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.835 + 0.549i)T \)
5 \( 1 + (0.802 - 0.597i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (-0.597 + 0.802i)T \)
13 \( 1 + (0.802 - 0.597i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.918 + 0.396i)T \)
31 \( 1 + (-0.957 - 0.286i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.835 + 0.549i)T \)
53 \( 1 + (0.893 - 0.448i)T \)
59 \( 1 + (-0.998 + 0.0581i)T \)
61 \( 1 + (-0.727 - 0.686i)T \)
67 \( 1 + (0.835 - 0.549i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.396 - 0.918i)T \)
79 \( 1 + (-0.998 + 0.0581i)T \)
83 \( 1 + (0.973 + 0.230i)T \)
89 \( 1 + (-0.116 - 0.993i)T \)
97 \( 1 + (-0.957 + 0.286i)T \)
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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

−18.39726857309496114703984513040, −18.18218963944982830733842413964, −17.647639254936652646186647984807, −16.39598503384662130014494529769, −15.50462001751677079649315207791, −14.81410397517150061844496289363, −13.86549326624193185377252974142, −13.77404139784193648007254423311, −13.16800858070330396339795274887, −12.22751643438060988441085241319, −11.50102248594858972527915037175, −11.07857087792161113847282236955, −9.98484158518757683168858556524, −9.51635235910506185736653458596, −8.69710941981855583376621698793, −8.368391141541740645530999804372, −7.3822391744428455059294117039, −6.386103837764726734809068156053, −5.696817705155158436949757392488, −4.940015071710442301708604713051, −3.75761038571902685180289450990, −3.06967220505290450508198608262, −2.53139144547786094988080998874, −1.77108939234095968599378685301, −1.23002945172748351390685221891, 0.155911790630650012425952591829, 1.28761584054874337950697249190, 2.07366820206355721370228483877, 3.28894051835989686542417139981, 4.04522421384730618620776165767, 4.85296991051477620726888941971, 5.136058965821308053589999615149, 6.22661062967981891191168515840, 7.0076853723150759818468830850, 7.84727841094667675900167589880, 8.33039663946236643576105649838, 8.97452427762142555269172049058, 9.77494188332734811078940381258, 10.28436576077652996134424763470, 10.86135851639127312316968631205, 12.38075125875390463009241106932, 13.13516239982527016311346727613, 13.52544602653640258455005951981, 14.08852098716854678061195507658, 14.76651838459773685479197612654, 15.51963971673989874471651973931, 16.040590581900908034644412236403, 16.69211112344436010878839423996, 17.38144740111601618062018397661, 18.10879451484933529857657954528

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