Properties

Label 1-4033-4033.2416-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.0144 - 0.999i$
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  + (0.342 − 0.939i)2-s + (0.396 + 0.918i)3-s + (−0.766 − 0.642i)4-s + (−0.973 + 0.230i)5-s + (0.998 − 0.0581i)6-s + (−0.286 + 0.957i)7-s + (−0.866 + 0.5i)8-s + (−0.686 + 0.727i)9-s + (−0.116 + 0.993i)10-s + (−0.116 − 0.993i)11-s + (0.286 − 0.957i)12-s + (0.957 + 0.286i)13-s + (0.802 + 0.597i)14-s + (−0.597 − 0.802i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.396 + 0.918i)3-s + (−0.766 − 0.642i)4-s + (−0.973 + 0.230i)5-s + (0.998 − 0.0581i)6-s + (−0.286 + 0.957i)7-s + (−0.866 + 0.5i)8-s + (−0.686 + 0.727i)9-s + (−0.116 + 0.993i)10-s + (−0.116 − 0.993i)11-s + (0.286 − 0.957i)12-s + (0.957 + 0.286i)13-s + (0.802 + 0.597i)14-s + (−0.597 − 0.802i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0144 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0144 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0144 - 0.999i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.0144 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4588380044 - 0.4655241200i\)
\(L(\frac12)\) \(\approx\) \(0.4588380044 - 0.4655241200i\)
\(L(1)\) \(\approx\) \(0.8988620866 - 0.04575140045i\)
\(L(1)\) \(\approx\) \(0.8988620866 - 0.04575140045i\)

Euler product

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

−18.19517432231346351149803250657, −17.86018476682596616390895949893, −17.10215783697733039534491820167, −16.33505878363993323576740041497, −15.61420725643979038140924984390, −15.21233990684754436309788885337, −14.34166164596847965823098399528, −13.58919697845258327398701164182, −13.19987050490563272719817497215, −12.611111962258470671145511907533, −11.743395403731469915376255981436, −11.23741207252910275278649606476, −9.8925732328277148802238167496, −9.27148924351103599732325382899, −8.26476851815187497516508547977, −7.948229064195079549139419399097, −7.1291901920696335290691244434, −6.89568641300478229172264950720, −5.9336884659835344675274341004, −5.04006224020224493288041670249, −4.03232313386109210734183487747, −3.723454177313394403498412420785, −2.80295957193760362519056898436, −1.5130989093614892271315190241, −0.56125517785232555923426464027, 0.1352676570253046820769049904, 1.34640045000861074544302556381, 2.57937362009233461803167730872, 3.03628391097364650566299428827, 3.68194892919785550674499862174, 4.34233837245793923296741017302, 5.14915845656560780083978402042, 5.85893614185550208249062530629, 6.70812533957347910935995388368, 8.09401275227731579095992751613, 8.7787416873130435342447109219, 8.90668872063974929333714012800, 9.93478629457358168424975300445, 10.8396261380667435526759391522, 11.134692078690525961824812359557, 11.73788743190720088721348924741, 12.592346393567100196796961780492, 13.28351036114356479510727506221, 14.240567047901631229702881608, 14.49993460361260685797210017974, 15.50653326704196448888702893328, 15.96748800105134907401155653966, 16.30459447710790401660319431833, 17.753370795237847509248229891719, 18.3889521150366341732009991373

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