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Properties

Label 1-3381-3381.1115-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.994 + 0.102i$
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  + (0.694 + 0.719i)2-s + (−0.0339 + 0.999i)4-s + (0.999 + 0.0271i)5-s + (−0.742 + 0.670i)8-s + (0.675 + 0.737i)10-s + (−0.546 − 0.837i)11-s + (0.685 − 0.728i)13-s + (−0.997 − 0.0679i)16-s + (−0.848 + 0.529i)17-s + (0.327 − 0.945i)19-s + (−0.0611 + 0.998i)20-s + (0.222 − 0.974i)22-s + (0.998 + 0.0543i)25-s + (0.999 − 0.0135i)26-s + (−0.882 − 0.470i)29-s + ⋯
L(s)  = 1  + (0.694 + 0.719i)2-s + (−0.0339 + 0.999i)4-s + (0.999 + 0.0271i)5-s + (−0.742 + 0.670i)8-s + (0.675 + 0.737i)10-s + (−0.546 − 0.837i)11-s + (0.685 − 0.728i)13-s + (−0.997 − 0.0679i)16-s + (−0.848 + 0.529i)17-s + (0.327 − 0.945i)19-s + (−0.0611 + 0.998i)20-s + (0.222 − 0.974i)22-s + (0.998 + 0.0543i)25-s + (0.999 − 0.0135i)26-s + (−0.882 − 0.470i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.994 + 0.102i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.994 + 0.102i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.689404458 + 0.1378958823i\)
\(L(\frac12)\) \(\approx\) \(2.689404458 + 0.1378958823i\)
\(L(1)\) \(\approx\) \(1.604618264 + 0.4672948598i\)
\(L(1)\) \(\approx\) \(1.604618264 + 0.4672948598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.694 + 0.719i)T \)
5 \( 1 + (0.999 + 0.0271i)T \)
11 \( 1 + (-0.546 - 0.837i)T \)
13 \( 1 + (0.685 - 0.728i)T \)
17 \( 1 + (-0.848 + 0.529i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.882 - 0.470i)T \)
31 \( 1 + (0.235 - 0.971i)T \)
37 \( 1 + (0.984 - 0.175i)T \)
41 \( 1 + (0.523 - 0.852i)T \)
43 \( 1 + (-0.742 - 0.670i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (0.777 - 0.628i)T \)
59 \( 1 + (0.675 + 0.737i)T \)
61 \( 1 + (-0.511 + 0.859i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
71 \( 1 + (-0.794 - 0.607i)T \)
73 \( 1 + (0.464 - 0.885i)T \)
79 \( 1 + (0.786 - 0.618i)T \)
83 \( 1 + (0.947 - 0.320i)T \)
89 \( 1 + (0.906 + 0.421i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−18.646231469629379510503077798740, −18.260798558937142944167668824595, −17.74931579325892958131354836927, −16.6027079101973278420271566563, −16.042543865350174202009719370739, −15.09203936601728418969407688878, −14.495427793894041584489190502458, −13.81683538343392388136311964917, −13.199566963264091430850373500701, −12.7290551728201585610095421955, −11.83648393698859758040834191823, −11.14597290679464475210641280009, −10.430926226191306862846917009812, −9.73111267521894418258685937206, −9.2695668735594993562580096897, −8.3477751877058198709467404025, −7.09193442863013349556160597725, −6.4868754227577794539457517396, −5.71870420035149694181341780829, −4.99763117687376251676267147997, −4.36281401928999681296222963234, −3.4030252686278263950049077615, −2.51703327472340381883732670959, −1.85272019935986760717326313553, −1.1740541485806781568999232788, 0.60315636624340492507878348485, 2.05596412040100301941042977222, 2.74715931980735623408177154352, 3.54675623614371914160046801555, 4.467860382287359768034845732745, 5.334724689667838268625604730735, 5.922424958913388530854138339115, 6.3653909234151715161323383548, 7.35237157531855687862333974935, 8.09662797890274087635561807496, 8.85279565565236259310326689576, 9.42940756701774018283918633826, 10.62725466855369934960666065894, 11.05665202881907793343530382073, 11.99029586890712924712713048066, 13.08763028293817125269415173380, 13.35965915762845773545649660547, 13.685535890197467725566820093968, 14.86487021602670834279673626691, 15.19081908938691530387479970944, 16.100601438772157252189013510881, 16.61613747030733513225905812412, 17.48683383012105620621792997879, 17.89410056374996150537564747163, 18.54254366241646269566118685053

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