Properties

Label 2-3381-1.1-c1-0-23
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  + 1.19·2-s − 3-s − 0.561·4-s − 0.572·5-s − 1.19·6-s − 3.07·8-s + 9-s − 0.686·10-s + 2.52·11-s + 0.561·12-s − 5.05·13-s + 0.572·15-s − 2.56·16-s − 2.25·17-s + 1.19·18-s + 2.39·19-s + 0.321·20-s + 3.02·22-s + 23-s + 3.07·24-s − 4.67·25-s − 6.05·26-s − 27-s + 0.796·29-s + 0.686·30-s − 3.45·31-s + 3.07·32-s + ⋯
L(s)  = 1  + 0.847·2-s − 0.577·3-s − 0.280·4-s − 0.256·5-s − 0.489·6-s − 1.08·8-s + 0.333·9-s − 0.217·10-s + 0.761·11-s + 0.162·12-s − 1.40·13-s + 0.147·15-s − 0.640·16-s − 0.546·17-s + 0.282·18-s + 0.550·19-s + 0.0719·20-s + 0.645·22-s + 0.208·23-s + 0.627·24-s − 0.934·25-s − 1.18·26-s − 0.192·27-s + 0.147·29-s + 0.125·30-s − 0.620·31-s + 0.543·32-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

\(L(1)\) \(\approx\) \(1.424374101\)
\(L(\frac12)\) \(\approx\) \(1.424374101\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.19T + 2T^{2} \)
5 \( 1 + 0.572T + 5T^{2} \)
11 \( 1 - 2.52T + 11T^{2} \)
13 \( 1 + 5.05T + 13T^{2} \)
17 \( 1 + 2.25T + 17T^{2} \)
19 \( 1 - 2.39T + 19T^{2} \)
29 \( 1 - 0.796T + 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 - 8.80T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 1.90T + 43T^{2} \)
47 \( 1 + 1.93T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 + 4.40T + 59T^{2} \)
61 \( 1 - 2.12T + 61T^{2} \)
67 \( 1 - 4.01T + 67T^{2} \)
71 \( 1 - 0.424T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 - 4.97T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 3.82T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−8.695483619163521405697756015486, −7.64324337865475392037650613867, −7.04373213231075224376753764234, −6.09022637779788071767256377248, −5.56797058856882452265645624046, −4.60566444690297872860095995291, −4.26160232791707291996634632338, −3.26462333428732712928114593185, −2.21992927247438352578935050670, −0.62854377213548230726438480507, 0.62854377213548230726438480507, 2.21992927247438352578935050670, 3.26462333428732712928114593185, 4.26160232791707291996634632338, 4.60566444690297872860095995291, 5.56797058856882452265645624046, 6.09022637779788071767256377248, 7.04373213231075224376753764234, 7.64324337865475392037650613867, 8.695483619163521405697756015486

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