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Properties

Label 1-4033-4033.675-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.865 - 0.500i$
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 + (0.597 − 0.802i)3-s + 4-s + (0.973 − 0.230i)5-s + (0.597 − 0.802i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.597 − 0.802i)12-s + (0.973 − 0.230i)13-s + (−0.286 + 0.957i)14-s + (0.396 − 0.918i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (0.597 − 0.802i)3-s + 4-s + (0.973 − 0.230i)5-s + (0.597 − 0.802i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.597 − 0.802i)12-s + (0.973 − 0.230i)13-s + (−0.286 + 0.957i)14-s + (0.396 − 0.918i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.865 - 0.500i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.865 - 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.827011238 - 1.564268818i\)
\(L(\frac12)\) \(\approx\) \(5.827011238 - 1.564268818i\)
\(L(1)\) \(\approx\) \(2.935398452 - 0.5738435593i\)
\(L(1)\) \(\approx\) \(2.935398452 - 0.5738435593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.597 - 0.802i)T \)
5 \( 1 + (0.973 - 0.230i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (0.973 + 0.230i)T \)
13 \( 1 + (0.973 - 0.230i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.286 + 0.957i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.597 + 0.802i)T \)
53 \( 1 + (-0.993 - 0.116i)T \)
59 \( 1 + (0.396 - 0.918i)T \)
61 \( 1 + (-0.835 + 0.549i)T \)
67 \( 1 + (0.597 + 0.802i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.286 - 0.957i)T \)
79 \( 1 + (0.396 - 0.918i)T \)
83 \( 1 + (-0.0581 - 0.998i)T \)
89 \( 1 + (-0.686 - 0.727i)T \)
97 \( 1 + (0.893 + 0.448i)T \)
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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

−18.76729399383324255807909284832, −17.49535523227999492363276833496, −16.96685577157487024986353658893, −16.44010837634861998921230996853, −15.56681802554090051569558190574, −15.13652163306856036777577444249, −14.11845125284703043875396338798, −13.75365207274794787972061402722, −13.56087121784216700831392213837, −12.63895962301598441601158119066, −11.3695097603976892436735999616, −11.092032811442281369366056285541, −10.28918995142936159361977775629, −9.64310915618757181599598700771, −8.96504195549347900131770466498, −8.05518564222733321659345333299, −6.96872810014352995005935978218, −6.55817036287414098682274587020, −5.74731303046731544179892873193, −4.89765867406775149212748360954, −4.1321654164192816062476259759, −3.63804303308642848636549592631, −2.85574722678479842077273779303, −2.03775192475124157537878368891, −1.18477719050654799384883344545, 1.24428948843365556600237383833, 1.80857149427441969908990579717, 2.46540368524870791780297890614, 3.238147882193363936613339077329, 4.05900830346873458911779242333, 4.96326607510611262481857074242, 6.089157942435750495952985797175, 6.25991598032059967297764223511, 6.72143714563045503957804289618, 8.0208709588679427777114034345, 8.64400752836514763796868024325, 9.203785200665607204082881687577, 10.18680025570638890729713525929, 11.00895295769617935966164861163, 11.88686403349907305914199510016, 12.57793532963212714148209572918, 12.946084802928791120031300981921, 13.486098460779728846606539030614, 14.407947375276708766206323088, 14.706864116302470628776386873199, 15.39848736927683203773525742376, 16.29579166885670004371534322071, 17.09439565843843627023618111658, 17.67242136863637555886689191506, 18.632718918936441266708989506319

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