Properties

Label 2-3381-1.1-c1-0-48
Degree $2$
Conductor $3381$
Sign $1$
Analytic cond. $26.9974$
Root an. cond. $5.19590$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0262·2-s − 3-s − 1.99·4-s + 2.77·5-s + 0.0262·6-s + 0.105·8-s + 9-s − 0.0729·10-s + 0.660·11-s + 1.99·12-s + 4.12·13-s − 2.77·15-s + 3.99·16-s + 4.12·17-s − 0.0262·18-s + 4.76·19-s − 5.55·20-s − 0.0173·22-s + 23-s − 0.105·24-s + 2.71·25-s − 0.108·26-s − 27-s − 1.78·29-s + 0.0729·30-s − 3.62·31-s − 0.314·32-s + ⋯
L(s)  = 1  − 0.0185·2-s − 0.577·3-s − 0.999·4-s + 1.24·5-s + 0.0107·6-s + 0.0371·8-s + 0.333·9-s − 0.0230·10-s + 0.199·11-s + 0.577·12-s + 1.14·13-s − 0.717·15-s + 0.998·16-s + 1.00·17-s − 0.00618·18-s + 1.09·19-s − 1.24·20-s − 0.00369·22-s + 0.208·23-s − 0.0214·24-s + 0.543·25-s − 0.0212·26-s − 0.192·27-s − 0.330·29-s + 0.0133·30-s − 0.650·31-s − 0.0556·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(26.9974\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.750623330\)
\(L(\frac12)\) \(\approx\) \(1.750623330\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
23 \( 1 - T \)
good2 \( 1 + 0.0262T + 2T^{2} \)
5 \( 1 - 2.77T + 5T^{2} \)
11 \( 1 - 0.660T + 11T^{2} \)
13 \( 1 - 4.12T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 - 4.76T + 19T^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 + 3.62T + 31T^{2} \)
37 \( 1 + 8.06T + 37T^{2} \)
41 \( 1 + 7.99T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + 0.0608T + 53T^{2} \)
59 \( 1 + 8.98T + 59T^{2} \)
61 \( 1 - 2.71T + 61T^{2} \)
67 \( 1 + 2.68T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 + 6.94T + 79T^{2} \)
83 \( 1 - 9.40T + 83T^{2} \)
89 \( 1 + 5.92T + 89T^{2} \)
97 \( 1 - 2.53T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.010827513387268161733065916686, −7.87548064614929192843968235075, −7.09975718114559717179457675395, −5.97904410390260699363481162515, −5.66794240894222689420427272054, −5.04016859417432559216449649379, −3.96226310274381039081756862836, −3.20540869236312593094745339289, −1.70328839329283785881970932437, −0.893261545656976000639346991188, 0.893261545656976000639346991188, 1.70328839329283785881970932437, 3.20540869236312593094745339289, 3.96226310274381039081756862836, 5.04016859417432559216449649379, 5.66794240894222689420427272054, 5.97904410390260699363481162515, 7.09975718114559717179457675395, 7.87548064614929192843968235075, 9.010827513387268161733065916686

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