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Properties

Label 1-4033-4033.591-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.443 + 0.896i$
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 + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.5 + 0.866i)15-s + 16-s + 17-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.5 + 0.866i)15-s + 16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.443 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.043174447 + 1.889210926i\)
\(L(\frac12)\) \(\approx\) \(3.043174447 + 1.889210926i\)
\(L(1)\) \(\approx\) \(1.875518590 + 0.5793917022i\)
\(L(1)\) \(\approx\) \(1.875518590 + 0.5793917022i\)

Euler product

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

−18.551723322082495406017096710139, −17.4960527019282002304687978105, −16.869863688011843522273421484379, −16.53737170749633403157749959991, −15.49897992645631771219105785671, −14.67208929520699885084378257554, −14.0743520937978890538220678441, −13.50152984188207747676701283442, −13.148464029922114560829010132772, −12.15981516324147512890617657478, −11.69370781791786646999162171761, −10.811106653307511995809675777746, −10.42649344141454383299037689705, −9.65154143338670710137351962848, −8.2285151697582713990552765133, −7.4576355487810893599371288384, −6.94286562771196418333573546521, −6.43856274946944381911392452022, −5.53215993179779293322583697218, −5.269278890553213058022167744902, −3.92292911008166969795309144239, −2.9974390119126910539364368957, −2.86267983413535731888707128489, −1.47740666907742194052618474393, −0.85605041115649121313181079754, 1.08220364063093281899878772523, 1.98977378072906928660855707482, 2.94740943018376387066976142976, 3.66485007131268950796383757241, 4.667239957761250332361332071624, 5.00311605278370022718044427478, 5.61795133421851590059430553329, 6.43051159453465130494623012705, 6.9813451305970757504040278980, 8.21530143441436803972276596338, 9.12882905963008364772076608959, 9.79216804348764379035931727437, 10.0763256403577489371141557314, 11.35164198788204701979164870324, 12.07060128436408572475438011880, 12.1499584693777059629368653104, 13.00321932757113567041844535321, 13.9265902864665012609721789039, 14.53550410922700997445188438039, 15.209230884365378071636536325823, 15.89707591238238570553040561604, 16.35553006421910293620220216392, 17.07834578997606445630005105921, 17.49218394647609035062133101051, 18.67181629431633897161131777273

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