Properties

Label 1-4033-4033.235-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.380 + 0.924i$
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 + (−0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.984 + 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.642 − 0.766i)10-s + (−0.642 − 0.766i)11-s − 12-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)14-s + (0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.984 + 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.642 − 0.766i)10-s + (−0.642 − 0.766i)11-s − 12-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)14-s + (0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.380 + 0.924i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.380 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5934759441 + 0.3975148652i\)
\(L(\frac12)\) \(\approx\) \(0.5934759441 + 0.3975148652i\)
\(L(1)\) \(\approx\) \(0.5877461416 + 0.08797792843i\)
\(L(1)\) \(\approx\) \(0.5877461416 + 0.08797792843i\)

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 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.642 - 0.766i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.342 - 0.939i)T \)
31 \( 1 + (0.342 - 0.939i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.984 + 0.173i)T \)
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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

−18.35617861415975305019234957664, −17.50009817994547081948997998044, −17.1554421823151688950930718566, −16.16560098700145099814538691422, −15.89069327643771971368326486176, −15.29394913450964720141619198161, −14.31331228149794360047281872133, −13.395351920640951349710961834417, −12.70601868161951748608503554285, −11.8955984743795184797069457041, −11.18477481139655608437059299385, −10.714847262074533017072139795019, −10.31938543099409374882072135617, −9.269474743386561564699980070518, −8.8244212095114777467924325558, −8.02545951767428501539984099940, −7.25682239699439731502300546444, −6.78897795943023254113926619263, −5.165956180504341006296178194626, −4.71168102820787652112196219188, −3.89221571117633231025103171250, −3.31684440072240855971069782812, −2.6020695271073021140212122961, −1.16180549005666011535336609645, −0.43270670256402469386163759299, 0.74972787004612896811064053875, 1.618707965566548129933386169334, 2.48917082040872972792486259713, 3.387830514293646856987022256115, 4.54800297184813458573039823846, 5.63716342926003788274801699268, 6.02703135533704023811178168805, 6.63564526678691117548643164310, 7.75644315563305052519385436969, 7.99714988854254803059227678193, 8.57238765442213725950770777896, 9.23279085101459308080668574083, 10.49968322155019754279470560924, 11.090280307864209988033664174, 11.576563808635368227851721205538, 12.34614421357199440212252940221, 13.1620564720495119405111212497, 13.88006090228969670788677289012, 14.84061971784485733420752826259, 15.15172212782103041773756083376, 16.018731458972602009483662458634, 16.55930819623183907149697992459, 17.26769017403387024922858434325, 18.105362593994375804551975768976, 18.79815505911776428765962852004

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