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Properties

Label 1-4033-4033.2871-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.968 + 0.247i$
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 + (0.286 − 0.957i)3-s + 4-s + (−0.448 + 0.893i)5-s + (−0.286 + 0.957i)6-s + (−0.835 + 0.549i)7-s − 8-s + (−0.835 − 0.549i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.286 − 0.957i)12-s + (−0.893 − 0.448i)13-s + (0.835 − 0.549i)14-s + (0.727 + 0.686i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.286 − 0.957i)3-s + 4-s + (−0.448 + 0.893i)5-s + (−0.286 + 0.957i)6-s + (−0.835 + 0.549i)7-s − 8-s + (−0.835 − 0.549i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.286 − 0.957i)12-s + (−0.893 − 0.448i)13-s + (0.835 − 0.549i)14-s + (0.727 + 0.686i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.968 + 0.247i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.968 + 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1253052305 + 0.01577920563i\)
\(L(\frac12)\) \(\approx\) \(0.1253052305 + 0.01577920563i\)
\(L(1)\) \(\approx\) \(0.3964078715 - 0.09829314990i\)
\(L(1)\) \(\approx\) \(0.3964078715 - 0.09829314990i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.286 - 0.957i)T \)
5 \( 1 + (-0.448 + 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.448 - 0.893i)T \)
13 \( 1 + (-0.893 - 0.448i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.549 - 0.835i)T \)
31 \( 1 + (-0.802 + 0.597i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (-0.957 + 0.286i)T \)
53 \( 1 + (-0.230 - 0.973i)T \)
59 \( 1 + (0.686 - 0.727i)T \)
61 \( 1 + (-0.918 + 0.396i)T \)
67 \( 1 + (-0.957 + 0.286i)T \)
71 \( 1 - T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (0.686 - 0.727i)T \)
83 \( 1 + (0.993 + 0.116i)T \)
89 \( 1 + (-0.998 + 0.0581i)T \)
97 \( 1 + (0.802 + 0.597i)T \)
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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

−18.44165105526077450320498637576, −17.63895613672014587660189683507, −16.7794928099628950296003079339, −16.41773052119630569738935610217, −16.11998606924138980366930436018, −15.10192390463618918749253416945, −14.823741525451648246747002964124, −13.68540450325511058711613272016, −12.75768632336538952429749819232, −12.20964104778067672751039284618, −11.389639107214088076051623247900, −10.62227377415527184038587338065, −9.90283758822201156454509897012, −9.48064320027324382210302066343, −8.96942347167833125122625141944, −8.00070278527576823951667847691, −7.54982402077157006264967066414, −6.69649705838754074442993970858, −5.74042820774071059763976527311, −4.76486724922414346331206419815, −4.2271640186842318675607212608, −3.3243798341019090507922844768, −2.44358413171415577106723790632, −1.644273989253002664843377225900, −0.12569261725571233396711421039, 0.32758314083764696786261558392, 1.9304340219364094498046824884, 2.38076839655046240187581336632, 3.135416045882418045137171308981, 3.731487046514705097121124684099, 5.51276874986579121350571305775, 6.15557100787929428105539615582, 6.700047900568061617194052489696, 7.37722935931939214739072023891, 8.07397840646709069972566590401, 8.59230880115691577710607547699, 9.38474396111755472732402678858, 10.22816316920026784784607925825, 10.874782119580944036751704920673, 11.57666797766903077226490699366, 12.24773651248975555019401567601, 12.888719051468533593284850546079, 13.63603865975318490233204689791, 14.70565714305820716196959627172, 15.105317359367566072672788835031, 15.78782278705714959216419435011, 16.53552815975709479231220768630, 17.47131241270162719336973841451, 17.88127107404297003821475027238, 18.63763805996717873955022579859

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