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Properties

Label 1-4577-4577.1025-r1-0-0
Degree $1$
Conductor $4577$
Sign $0.658 + 0.752i$
Analytic cond. $491.866$
Root an. cond. $491.866$
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  + (−0.916 + 0.400i)2-s + (−0.873 + 0.486i)3-s + (0.678 − 0.734i)4-s + (−0.786 + 0.618i)5-s + (0.605 − 0.795i)6-s + (0.296 − 0.954i)7-s + (−0.327 + 0.945i)8-s + (0.527 − 0.849i)9-s + (0.472 − 0.881i)10-s + (−0.841 − 0.540i)11-s + (−0.235 + 0.971i)12-s + (−0.0792 − 0.996i)13-s + (0.110 + 0.993i)14-s + (0.386 − 0.922i)15-s + (−0.0792 − 0.996i)16-s + (−0.580 − 0.814i)17-s + ⋯
L(s)  = 1  + (−0.916 + 0.400i)2-s + (−0.873 + 0.486i)3-s + (0.678 − 0.734i)4-s + (−0.786 + 0.618i)5-s + (0.605 − 0.795i)6-s + (0.296 − 0.954i)7-s + (−0.327 + 0.945i)8-s + (0.527 − 0.849i)9-s + (0.472 − 0.881i)10-s + (−0.841 − 0.540i)11-s + (−0.235 + 0.971i)12-s + (−0.0792 − 0.996i)13-s + (0.110 + 0.993i)14-s + (0.386 − 0.922i)15-s + (−0.0792 − 0.996i)16-s + (−0.580 − 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4577 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4577 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4577\)    =    \(23 \cdot 199\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(491.866\)
Root analytic conductor: \(491.866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4577} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4577,\ (1:\ ),\ 0.658 + 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3766532750 + 0.1709241982i\)
\(L(\frac12)\) \(\approx\) \(0.3766532750 + 0.1709241982i\)
\(L(1)\) \(\approx\) \(0.4286582572 + 0.03139298200i\)
\(L(1)\) \(\approx\) \(0.4286582572 + 0.03139298200i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
199 \( 1 \)
good2 \( 1 + (-0.916 + 0.400i)T \)
3 \( 1 + (-0.873 + 0.486i)T \)
5 \( 1 + (-0.786 + 0.618i)T \)
7 \( 1 + (0.296 - 0.954i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (-0.0792 - 0.996i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.805 - 0.592i)T \)
29 \( 1 + (0.902 + 0.429i)T \)
31 \( 1 + (-0.0158 - 0.999i)T \)
37 \( 1 + (0.987 - 0.158i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.902 + 0.429i)T \)
47 \( 1 + (0.967 + 0.251i)T \)
53 \( 1 + (-0.823 - 0.567i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
71 \( 1 + (0.823 - 0.567i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.327 - 0.945i)T \)
89 \( 1 + (-0.975 + 0.220i)T \)
97 \( 1 + (-0.110 - 0.993i)T \)
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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

−17.993909191024665680499604475621, −17.34398650698042073137141279158, −16.8382394319829254351822622312, −16.00049628751685826564074741040, −15.65258685053725179447980601710, −14.939171158454958422810882501061, −13.67188282967975415192743847184, −12.665585522453594156472680321578, −12.377412184380312985959398979719, −11.986324922293045059223680076893, −11.05917682560327274821327715888, −10.74850967840354074407772989538, −9.7862854461979428847358528307, −8.92166062165242312538543775902, −8.31538939313396050127490205254, −7.814101418680686108052156754714, −6.97581144680113879577735806889, −6.31395344875415935538117237715, −5.44359306380380305694191874572, −4.5454063959602079855739859340, −3.988929235622183980944762408958, −2.5793844238461702774987949059, −1.98977969645130855346944347460, −1.27976427663269465258125244124, −0.21799532256824524405762654473, 0.49092162537619643833896979359, 0.90257820616709970647281968162, 2.47452282558640696460729516253, 3.145842815336705935722191449840, 4.29886746071336581610794710276, 4.83530835903246044066666038045, 5.762597754157362500535758279495, 6.49335934184417646192070522055, 7.128487114894901588370671298208, 7.791230611208479594116051525282, 8.32927362204340944744928902309, 9.3780674415447898594676087772, 10.155272951093666457310592903770, 10.71522401968207471227617463271, 11.08877224663001318695678215948, 11.56556832688783327768652287387, 12.62501081584439941629325628781, 13.45599920733172887537819205802, 14.459181442447314943102321667345, 15.07121975730021872197365742784, 15.69770679939548383603983609590, 16.128922868090831710181314422648, 16.781649491609566148066745361440, 17.62924489170002043504945275009, 17.90872924562429829296221709823

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