Properties

Label 2-3381-1.1-c1-0-36
Degree $2$
Conductor $3381$
Sign $1$
Analytic cond. $26.9974$
Root an. cond. $5.19590$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 4·11-s + 12-s + 6·13-s − 16-s − 4·17-s + 18-s − 2·19-s + 4·22-s − 23-s + 3·24-s − 5·25-s + 6·26-s − 27-s + 2·29-s − 4·31-s + 5·32-s − 4·33-s − 4·34-s − 36-s + 2·37-s − 2·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1.66·13-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.458·19-s + 0.852·22-s − 0.208·23-s + 0.612·24-s − 25-s + 1.17·26-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.883·32-s − 0.696·33-s − 0.685·34-s − 1/6·36-s + 0.328·37-s − 0.324·38-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.804380550\)
\(L(\frac12)\) \(\approx\) \(1.804380550\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.777189075877992884842439260940, −7.932064828839316884026498283002, −6.70206404150808998760951214918, −6.23021103057897426141874753360, −5.68195104737116298433568694308, −4.64983466386812914028477867317, −3.99318231398277204106985985079, −3.51073144451088399010587234214, −2.00022392271948453847295980974, −0.75739838906060291586223443436, 0.75739838906060291586223443436, 2.00022392271948453847295980974, 3.51073144451088399010587234214, 3.99318231398277204106985985079, 4.64983466386812914028477867317, 5.68195104737116298433568694308, 6.23021103057897426141874753360, 6.70206404150808998760951214918, 7.932064828839316884026498283002, 8.777189075877992884842439260940

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