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Properties

Label 1-3100-3100.487-r1-0-0
Degree $1$
Conductor $3100$
Sign $-0.469 - 0.882i$
Analytic cond. $333.141$
Root an. cond. $333.141$
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  + (0.207 + 0.978i)3-s + (−0.743 − 0.669i)7-s + (−0.913 + 0.406i)9-s + (−0.104 − 0.994i)11-s + (0.207 − 0.978i)13-s + (0.406 − 0.913i)17-s + (0.669 − 0.743i)19-s + (0.5 − 0.866i)21-s − i·23-s + (−0.587 − 0.809i)27-s + 29-s + (0.951 − 0.309i)33-s + (0.207 + 0.978i)37-s + 39-s + (−0.104 − 0.994i)41-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)3-s + (−0.743 − 0.669i)7-s + (−0.913 + 0.406i)9-s + (−0.104 − 0.994i)11-s + (0.207 − 0.978i)13-s + (0.406 − 0.913i)17-s + (0.669 − 0.743i)19-s + (0.5 − 0.866i)21-s − i·23-s + (−0.587 − 0.809i)27-s + 29-s + (0.951 − 0.309i)33-s + (0.207 + 0.978i)37-s + 39-s + (−0.104 − 0.994i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $-0.469 - 0.882i$
Analytic conductor: \(333.141\)
Root analytic conductor: \(333.141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3100} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3100,\ (1:\ ),\ -0.469 - 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9117566968 - 1.518038488i\)
\(L(\frac12)\) \(\approx\) \(0.9117566968 - 1.518038488i\)
\(L(1)\) \(\approx\) \(1.034155861 - 0.1055556587i\)
\(L(1)\) \(\approx\) \(1.034155861 - 0.1055556587i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.207 + 0.978i)T \)
7 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.406 - 0.913i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 - iT \)
29 \( 1 + T \)
37 \( 1 + (0.207 + 0.978i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.994 + 0.104i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−19.18565288877139161983673968160, −18.318085446469538362462441340991, −17.88637331331218004094773781290, −17.059408366979848102222889191993, −16.27989435337590040532367915291, −15.5645478487075275387100358163, −14.70192715223569434546714752114, −14.19709005422406593929459347287, −13.281880817151732869569269910483, −12.75487050996115768213496732020, −12.02140026582925455355681882168, −11.69607400026455164531306466648, −10.49827979151265808477225375716, −9.619015592749146906519655192580, −9.11556800578945173666471018416, −8.23122089334466944593698817407, −7.507217979302291194184371095114, −6.82930034839234309331222457046, −6.061467393806465394570744572418, −5.53852819485146004215904840244, −4.31382685838705186803055603665, −3.4351236991425757820876369416, −2.60523126863332652245361661440, −1.800024706574458863413166098994, −1.11186870416693381794908977964, 0.35759004137968306402411061209, 0.7973079553822414666401104175, 2.58954875085289569785121415721, 3.13792937463158808637725277319, 3.69610351933947810496831552727, 4.74437357136650133872443142258, 5.334647458294924333806917608762, 6.21764617934632619307056072862, 7.01369813635013570676216061628, 8.06095509655071352643320186346, 8.55453618684840669458587776565, 9.56217333848046050012566371868, 9.96393098285841878922595221687, 10.79640796694396808410080950394, 11.23549599768526864443243541179, 12.25515582133924009918562569274, 13.13474096055524359662185903708, 13.899684009201687663153599287387, 14.20230811365373045843850593110, 15.37419734548292851863114741579, 15.82043240418634951782752615388, 16.41346209928503651116911925553, 16.96908510950885931233846879532, 17.8460310911199563329488518509, 18.64310117362171967468616462661

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