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Properties

Label 1-4029-4029.491-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.889 - 0.456i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.464 − 0.885i)2-s + (−0.568 − 0.822i)4-s + (0.911 + 0.410i)5-s + (−0.787 + 0.616i)7-s + (−0.992 + 0.120i)8-s + (0.787 − 0.616i)10-s + (−0.911 + 0.410i)11-s + (−0.568 − 0.822i)13-s + (0.180 + 0.983i)14-s + (−0.354 + 0.935i)16-s + (0.663 − 0.748i)19-s + (−0.180 − 0.983i)20-s + (−0.0603 + 0.998i)22-s + (0.707 + 0.707i)23-s + (0.663 + 0.748i)25-s + (−0.992 + 0.120i)26-s + ⋯
L(s)  = 1  + (0.464 − 0.885i)2-s + (−0.568 − 0.822i)4-s + (0.911 + 0.410i)5-s + (−0.787 + 0.616i)7-s + (−0.992 + 0.120i)8-s + (0.787 − 0.616i)10-s + (−0.911 + 0.410i)11-s + (−0.568 − 0.822i)13-s + (0.180 + 0.983i)14-s + (−0.354 + 0.935i)16-s + (0.663 − 0.748i)19-s + (−0.180 − 0.983i)20-s + (−0.0603 + 0.998i)22-s + (0.707 + 0.707i)23-s + (0.663 + 0.748i)25-s + (−0.992 + 0.120i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.889 - 0.456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2681813653 - 1.109139086i\)
\(L(\frac12)\) \(\approx\) \(0.2681813653 - 1.109139086i\)
\(L(1)\) \(\approx\) \(0.9740372036 - 0.4950042289i\)
\(L(1)\) \(\approx\) \(0.9740372036 - 0.4950042289i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.464 - 0.885i)T \)
5 \( 1 + (0.911 + 0.410i)T \)
7 \( 1 + (-0.787 + 0.616i)T \)
11 \( 1 + (-0.911 + 0.410i)T \)
13 \( 1 + (-0.568 - 0.822i)T \)
19 \( 1 + (0.663 - 0.748i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.517 + 0.855i)T \)
31 \( 1 + (-0.954 + 0.297i)T \)
37 \( 1 + (-0.998 + 0.0603i)T \)
41 \( 1 + (0.410 - 0.911i)T \)
43 \( 1 + (0.935 - 0.354i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (-0.992 + 0.120i)T \)
59 \( 1 + (-0.822 - 0.568i)T \)
61 \( 1 + (0.0603 + 0.998i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (-0.787 - 0.616i)T \)
73 \( 1 + (-0.180 - 0.983i)T \)
83 \( 1 + (0.822 - 0.568i)T \)
89 \( 1 + (0.120 - 0.992i)T \)
97 \( 1 + (-0.998 + 0.0603i)T \)
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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

−18.69873433953632635098279927798, −17.87982295066467461566670125126, −17.11832434550609277881835780570, −16.66560881341385297864929193905, −16.170405925217683603220725380214, −15.51102506837232310090293199745, −14.39740003589798995298345742497, −14.10607393300700869955925979822, −13.3490723078074889802237992717, −12.81828956439396370504670164251, −12.331827589637896182952221674604, −11.20527533827900147473523308529, −10.32194056944742793556460874195, −9.49054296570621812808558260910, −9.194664908855696775120194930160, −8.13314056856947254099993277089, −7.47546013981998211223433916819, −6.7398893922416320444426028312, −6.03582143895104427784462886358, −5.45513090522604911031774750923, −4.68855289571911826502023606716, −3.93943740199485354979458647506, −3.01545258577987838663167289696, −2.290971052412026216866502398775, −0.964282370469494417329880451417, 0.28949791989948824046007016421, 1.617374228214551750041595192935, 2.34854500227482811488578254068, 3.01931858759997882091883316662, 3.45116984286014401292342821075, 4.8728622763861443089086099292, 5.40884826312291554746912482775, 5.79785917209973671922388480063, 6.90015998631396050272379282972, 7.506495731858015972671498253499, 9.00717886035573624330657114186, 9.19595794923170897050845635231, 10.06438579197566568878734224728, 10.56214191545264945483455784251, 11.17198828934781461665485212912, 12.29774277731103912879997914164, 12.66019205597140349975064721291, 13.29852619318816306291057140164, 13.86629636296958636972567363759, 14.72488597341277768672831963511, 15.3484256772279105064145562865, 15.847828974792732601316853890254, 17.05495829880237800832089754235, 17.83230278302227983654569583315, 18.16875821491078437113390969106

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