Properties

Label 2-3381-1.1-c1-0-2
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  + 0.864·2-s − 3-s − 1.25·4-s − 2.49·5-s − 0.864·6-s − 2.81·8-s + 9-s − 2.15·10-s − 3.98·11-s + 1.25·12-s + 2.80·13-s + 2.49·15-s + 0.0715·16-s − 6.98·17-s + 0.864·18-s − 5.98·19-s + 3.12·20-s − 3.44·22-s + 23-s + 2.81·24-s + 1.22·25-s + 2.42·26-s − 27-s − 4.06·29-s + 2.15·30-s − 3.44·31-s + 5.68·32-s + ⋯
L(s)  = 1  + 0.611·2-s − 0.577·3-s − 0.626·4-s − 1.11·5-s − 0.353·6-s − 0.994·8-s + 0.333·9-s − 0.682·10-s − 1.20·11-s + 0.361·12-s + 0.778·13-s + 0.644·15-s + 0.0178·16-s − 1.69·17-s + 0.203·18-s − 1.37·19-s + 0.698·20-s − 0.734·22-s + 0.208·23-s + 0.574·24-s + 0.245·25-s + 0.476·26-s − 0.192·27-s − 0.754·29-s + 0.394·30-s − 0.618·31-s + 1.00·32-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

\(L(1)\) \(\approx\) \(0.3613976341\)
\(L(\frac12)\) \(\approx\) \(0.3613976341\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.864T + 2T^{2} \)
5 \( 1 + 2.49T + 5T^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 - 2.80T + 13T^{2} \)
17 \( 1 + 6.98T + 17T^{2} \)
19 \( 1 + 5.98T + 19T^{2} \)
29 \( 1 + 4.06T + 29T^{2} \)
31 \( 1 + 3.44T + 31T^{2} \)
37 \( 1 + 1.47T + 37T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 + 3.04T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 + 8.30T + 59T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 - 4.46T + 67T^{2} \)
71 \( 1 + 9.49T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 - 1.05T + 83T^{2} \)
89 \( 1 + 3.30T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−8.576863529985623865308807453016, −7.934617500702618849461115489815, −7.01999286371079562891771902586, −6.23010622559826214827122516918, −5.45889339350092336025635898166, −4.63602004976420992210990347715, −4.13624463216446413779197050825, −3.38619522660921725918986420073, −2.16501976726950543946951551575, −0.31783034320438359111323186294, 0.31783034320438359111323186294, 2.16501976726950543946951551575, 3.38619522660921725918986420073, 4.13624463216446413779197050825, 4.63602004976420992210990347715, 5.45889339350092336025635898166, 6.23010622559826214827122516918, 7.01999286371079562891771902586, 7.934617500702618849461115489815, 8.576863529985623865308807453016

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