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Properties

Label 1-4033-4033.1006-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0560 + 0.998i$
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  + (0.766 + 0.642i)2-s + (−0.286 + 0.957i)3-s + (0.173 + 0.984i)4-s + (−0.835 + 0.549i)5-s + (−0.835 + 0.549i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (−0.993 − 0.116i)12-s + (0.893 + 0.448i)13-s + (0.597 − 0.802i)14-s + (−0.286 − 0.957i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.286 + 0.957i)3-s + (0.173 + 0.984i)4-s + (−0.835 + 0.549i)5-s + (−0.835 + 0.549i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (−0.993 − 0.116i)12-s + (0.893 + 0.448i)13-s + (0.597 − 0.802i)14-s + (−0.286 − 0.957i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0560 + 0.998i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1006, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0560 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074803073 + 1.016109948i\)
\(L(\frac12)\) \(\approx\) \(1.074803073 + 1.016109948i\)
\(L(1)\) \(\approx\) \(0.8651187175 + 0.6977302625i\)
\(L(1)\) \(\approx\) \(0.8651187175 + 0.6977302625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (-0.286 + 0.957i)T \)
5 \( 1 + (-0.835 + 0.549i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (-0.993 + 0.116i)T \)
13 \( 1 + (0.893 + 0.448i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.993 - 0.116i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (-0.993 - 0.116i)T \)
59 \( 1 + (-0.286 - 0.957i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.597 + 0.802i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (0.396 - 0.918i)T \)
83 \( 1 + (-0.0581 - 0.998i)T \)
89 \( 1 + (0.973 - 0.230i)T \)
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

−18.57240097170729828191819324373, −18.027467846684322188887568652759, −16.86763805320059705799597432576, −16.01395342965481348684167922885, −15.613089802443934016210194661568, −14.845880261883946659285617437673, −13.99201390307691668979267008860, −13.2947819257546155352535280346, −12.53985908617732983153837555758, −12.44547190560915806413003240423, −11.55137484870179366588995536040, −11.10651825651663898389398512821, −10.30979684991808120842928831544, −9.205670900192733657725826988254, −8.50879370109106844028337856629, −7.79247590947640527487602990118, −7.07743141403147917801978144505, −6.003685105348159268702956469418, −5.37722023650191022180398375240, −5.182727081771659952458913913085, −3.77830772836598712085335077648, −3.234658106655306520855639596016, −2.34411671076679827784694460107, −1.54850859083336576292352153235, −0.68556182996754688414492509592, 0.4919469960582415338061302251, 2.30876362228922812542611388532, 3.32429496525364340353464600562, 3.6606229454596186378801001913, 4.535113450512255920886900749557, 4.833186869746535001442278205320, 6.18295383331743717392979497431, 6.34272744409202634527614226950, 7.54236625172186191755847629439, 7.893100094614538740538857517718, 8.74529472580596989515117287931, 9.7080301410995014211288704467, 10.715144188735676691122914806631, 11.02503098031034492180670973325, 11.57088339529821773129760947310, 12.760176283357528652227687510000, 13.07352899750434164877755348124, 14.24739317217149132415490752199, 14.543138215504535694632372978478, 15.31294048054706229110432262825, 15.87145743875214347900960463563, 16.39761548468311902073224321806, 16.960217135570684694285380907137, 17.70076193635046755874086635768, 18.53576704917140321238149468683

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