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Properties

Label 1-4033-4033.2253-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.472 - 0.881i$
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 + (−0.835 + 0.549i)3-s + (0.173 + 0.984i)4-s + (0.396 + 0.918i)5-s + (−0.993 − 0.116i)6-s + (−0.993 − 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.686 − 0.727i)12-s + (0.597 − 0.802i)13-s + (−0.686 − 0.727i)14-s + (−0.835 − 0.549i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.835 + 0.549i)3-s + (0.173 + 0.984i)4-s + (0.396 + 0.918i)5-s + (−0.993 − 0.116i)6-s + (−0.993 − 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.686 − 0.727i)12-s + (0.597 − 0.802i)13-s + (−0.686 − 0.727i)14-s + (−0.835 − 0.549i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.472 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2643433080 + 0.1581894142i\)
\(L(\frac12)\) \(\approx\) \(-0.2643433080 + 0.1581894142i\)
\(L(1)\) \(\approx\) \(0.6638131724 + 0.6824246537i\)
\(L(1)\) \(\approx\) \(0.6638131724 + 0.6824246537i\)

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 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (0.396 + 0.918i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.286 + 0.957i)T \)
31 \( 1 + (-0.0581 + 0.998i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.597 + 0.802i)T \)
53 \( 1 + (-0.686 - 0.727i)T \)
59 \( 1 + (-0.835 - 0.549i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (0.973 + 0.230i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (-0.0581 - 0.998i)T \)
97 \( 1 + (-0.835 + 0.549i)T \)
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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

−18.19109094168497926879937302572, −16.98979066854192926703717839285, −16.72038274411436052841636432683, −15.716073245988080910337639162429, −15.44271653862732105223247179552, −14.07345103154476262726898187081, −13.54017201318663387194325135275, −13.04152973063456886910334928549, −12.39720390713571703957914070757, −11.97282605316111843609527252456, −11.29272308299144960255482452653, −10.316395313546529964719011728319, −9.747105687758406762326661060125, −9.233086450929634104411892461555, −8.071297487570756368066011386, −7.03421131136038173140752453952, −6.39578188151568569618584501525, −5.84922127739885028763547805523, −5.07100740871252784842113218571, −4.47305092173746725466073429345, −3.71842913452319080277139603205, −2.38052254207589600398792915211, −2.00316065273316478022004160286, −0.982139511412462243420764215325, −0.07874423058597604358050983946, 1.556985162322313994303635004, 2.98424509163619676460223191950, 3.48462705445200527818648825844, 3.84055546653078326255969356027, 5.22910961206361553893521022636, 5.69229733669671143287517954264, 6.301851381501486466201034942575, 6.66333098695874159596158620690, 7.713855192930281275074043183226, 8.44707723022284159078748404100, 9.46582455858046547674974117890, 10.405210848376021053648159314498, 10.60829596151431024747390717358, 11.510357450585608123469369730601, 12.37510999508238924642055112836, 12.85696741958767003995395290672, 13.678002567856618714345611250862, 14.30615405165377409333573283486, 15.05825599105633982254392363073, 15.69357177225186841801010267410, 16.23657445833830769372003308459, 16.80047716441397656582937341955, 17.44196643584640017006626912196, 18.31262454432451753814826332137, 18.672926328400085592572762882785

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