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Properties

Label 1-4033-4033.3159-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.944 + 0.328i$
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 + (−0.173 + 0.984i)3-s + 4-s + (0.342 + 0.939i)5-s + (0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s − 8-s + (−0.939 − 0.342i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.939 − 0.342i)14-s + (−0.984 + 0.173i)15-s + 16-s + 17-s + ⋯
L(s)  = 1  − 2-s + (−0.173 + 0.984i)3-s + 4-s + (0.342 + 0.939i)5-s + (0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s − 8-s + (−0.939 − 0.342i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.939 − 0.342i)14-s + (−0.984 + 0.173i)15-s + 16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.944 + 0.328i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8272996694 + 0.1398814339i\)
\(L(\frac12)\) \(\approx\) \(0.8272996694 + 0.1398814339i\)
\(L(1)\) \(\approx\) \(0.6162365547 + 0.2342605315i\)
\(L(1)\) \(\approx\) \(0.6162365547 + 0.2342605315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (0.342 + 0.939i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.342 - 0.939i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.342 + 0.939i)T \)
31 \( 1 + (-0.642 - 0.766i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (0.984 + 0.173i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 - T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
97 \( 1 + (0.642 - 0.766i)T \)
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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

−18.206144715172062533087450927543, −17.84373384061717978162723002898, −17.19720609939297840376587173693, −16.471070224070675736281391431966, −16.15473157440127837124127536485, −15.26194917559919622324183955329, −14.212108748870907915158511046012, −13.41544576768243219196765054540, −12.96002500823412760966928799649, −11.97997232437608300488901209467, −11.88867426620114791045011238906, −10.781923212254193823534162675, −9.893384126546519999313839095331, −9.40751619081578606840727630685, −8.74076028243121106019925433373, −7.95889889842080127084361857253, −7.263475500457032105661804011984, −6.67422326434860325071695655292, −5.94101780622572931924104340907, −5.33118632859774322502178963141, −4.026363944070308201787638616670, −3.09181262487451804330711203335, −2.14634224146966124140515398917, −1.34048211054558058942915311901, −0.856999869604893990588967597944, 0.436328717365307173483520514657, 1.64601805397112554182020289562, 2.86247741420288192066642270376, 3.281719047607891040051326123963, 3.79460118932797721199420502905, 5.45040989546608593663937683671, 6.05197101371817573293766582424, 6.30566254914472682870303656161, 7.36800774770581842683500084799, 8.33189848324827872440201645277, 8.845696176001105239862403889663, 9.76539065906908036495013602064, 10.05375892137568802941006741076, 10.73425623137216014581895075991, 11.35053712456879678453773880341, 11.997186822851308961673803460830, 12.93331280800664513592161271163, 14.08018204393766846000575242592, 14.55904985606700519408223515412, 15.285540544514389663311843394689, 16.16327990804209046330064460140, 16.31798754757831517665782643662, 16.97789526643824931910251132756, 18.002100113834496988549544147333, 18.484255670414220955189975684799

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