Properties

Label 2-3381-1.1-c1-0-5
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  − 0.769·2-s − 3-s − 1.40·4-s − 4.02·5-s + 0.769·6-s + 2.62·8-s + 9-s + 3.09·10-s + 5.47·11-s + 1.40·12-s − 4.39·13-s + 4.02·15-s + 0.800·16-s − 2.24·17-s − 0.769·18-s + 3.56·19-s + 5.67·20-s − 4.20·22-s + 23-s − 2.62·24-s + 11.2·25-s + 3.38·26-s − 27-s − 4.70·29-s − 3.09·30-s − 10.5·31-s − 5.85·32-s + ⋯
L(s)  = 1  − 0.543·2-s − 0.577·3-s − 0.704·4-s − 1.80·5-s + 0.313·6-s + 0.926·8-s + 0.333·9-s + 0.979·10-s + 1.64·11-s + 0.406·12-s − 1.22·13-s + 1.03·15-s + 0.200·16-s − 0.545·17-s − 0.181·18-s + 0.817·19-s + 1.26·20-s − 0.897·22-s + 0.208·23-s − 0.535·24-s + 2.24·25-s + 0.663·26-s − 0.192·27-s − 0.874·29-s − 0.565·30-s − 1.89·31-s − 1.03·32-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

\(L(1)\) \(\approx\) \(0.3288182399\)
\(L(\frac12)\) \(\approx\) \(0.3288182399\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.769T + 2T^{2} \)
5 \( 1 + 4.02T + 5T^{2} \)
11 \( 1 - 5.47T + 11T^{2} \)
13 \( 1 + 4.39T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 - 3.56T + 19T^{2} \)
29 \( 1 + 4.70T + 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 3.92T + 37T^{2} \)
41 \( 1 + 3.97T + 41T^{2} \)
43 \( 1 - 2.61T + 43T^{2} \)
47 \( 1 + 0.518T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 0.842T + 59T^{2} \)
61 \( 1 - 9.00T + 61T^{2} \)
67 \( 1 + 1.39T + 67T^{2} \)
71 \( 1 + 7.10T + 71T^{2} \)
73 \( 1 - 5.32T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 3.39T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 6.35T + 97T^{2} \)
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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

−8.748965965705160338522225594028, −7.72788428873372315512130154153, −7.32621846913093444427197016096, −6.71171986323785627472061122160, −5.36986426979709998190398631783, −4.67321399587649656146673200795, −3.98195590470863281318047229244, −3.42263851918725971075452601383, −1.58217343181722003791338663778, −0.39988791219298933628040433637, 0.39988791219298933628040433637, 1.58217343181722003791338663778, 3.42263851918725971075452601383, 3.98195590470863281318047229244, 4.67321399587649656146673200795, 5.36986426979709998190398631783, 6.71171986323785627472061122160, 7.32621846913093444427197016096, 7.72788428873372315512130154153, 8.748965965705160338522225594028

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