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Properties

Label 1-3381-3381.1262-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.929 - 0.368i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.0882 − 0.996i)2-s + (−0.984 − 0.175i)4-s + (0.440 − 0.897i)5-s + (−0.262 + 0.965i)8-s + (−0.855 − 0.517i)10-s + (0.990 + 0.135i)11-s + (−0.452 + 0.891i)13-s + (0.938 + 0.346i)16-s + (0.644 + 0.764i)17-s + (0.888 + 0.458i)19-s + (−0.591 + 0.806i)20-s + (0.222 − 0.974i)22-s + (−0.612 − 0.790i)25-s + (0.848 + 0.529i)26-s + (−0.339 + 0.940i)29-s + ⋯
L(s)  = 1  + (0.0882 − 0.996i)2-s + (−0.984 − 0.175i)4-s + (0.440 − 0.897i)5-s + (−0.262 + 0.965i)8-s + (−0.855 − 0.517i)10-s + (0.990 + 0.135i)11-s + (−0.452 + 0.891i)13-s + (0.938 + 0.346i)16-s + (0.644 + 0.764i)17-s + (0.888 + 0.458i)19-s + (−0.591 + 0.806i)20-s + (0.222 − 0.974i)22-s + (−0.612 − 0.790i)25-s + (0.848 + 0.529i)26-s + (−0.339 + 0.940i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.584229155 - 0.3028548790i\)
\(L(\frac12)\) \(\approx\) \(1.584229155 - 0.3028548790i\)
\(L(1)\) \(\approx\) \(1.007639168 - 0.4792013898i\)
\(L(1)\) \(\approx\) \(1.007639168 - 0.4792013898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0882 - 0.996i)T \)
5 \( 1 + (0.440 - 0.897i)T \)
11 \( 1 + (0.990 + 0.135i)T \)
13 \( 1 + (-0.452 + 0.891i)T \)
17 \( 1 + (0.644 + 0.764i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.339 + 0.940i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (-0.557 - 0.830i)T \)
43 \( 1 + (-0.262 - 0.965i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (-0.923 + 0.384i)T \)
59 \( 1 + (-0.855 - 0.517i)T \)
61 \( 1 + (0.0339 + 0.999i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.0611 + 0.998i)T \)
73 \( 1 + (0.810 + 0.585i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.685 + 0.728i)T \)
89 \( 1 + (0.534 + 0.844i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−18.5471698660614452093477544459, −18.0524565952877512009078253951, −17.506008660670286523644119132685, −16.774522187336874852982973794555, −16.12866110276489401879133268324, −15.246753728869548576365836397781, −14.76850427763944598016012665392, −14.10711301639756978293799398014, −13.62747395409882120137579482610, −12.789368567315242532058666476682, −11.922434272205572746967540443316, −11.18945143932623282403463608976, −10.14839729591879482567368455087, −9.60521077565553208171550413482, −9.0515228507349587216424810321, −7.86632019209230045020121669481, −7.480909148679243755728023141461, −6.64570503606622242475176183591, −6.0789211254930189787312470103, −5.29540372371942581819476413568, −4.60605854890289525744508014453, −3.32631382496129660233242405944, −3.13256536900348145846868845227, −1.70653792506259630953401234861, −0.51134509256283687315366298476, 1.114496045725775215723461078846, 1.541768539022593973390330588750, 2.392867962667490420635546621, 3.5664240357466363630026819297, 4.097415762230107589286398605274, 4.93779319112355426523171952224, 5.60026502029585682337572386036, 6.41738661643119917791397447096, 7.55620867135285758541256709278, 8.422099367137305154646498605814, 9.14785741754434509379702009707, 9.60912124934776481766581385067, 10.21611934267065642195333214956, 11.24920919539788810342689663588, 11.90787665624498561051736849110, 12.39195832587252043312249509771, 13.03443438861417306293448736862, 13.93895011794094276478751003578, 14.29117144992982559415058081845, 15.10087473453675878057058181808, 16.29140070776536884346570228813, 16.976753605791891401157537402355, 17.252361873762283687470259566041, 18.25611641799455730435358168723, 18.92035338516509331656566587154

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