Properties

Label 2-3330-1.1-c1-0-9
Degree $2$
Conductor $3330$
Sign $1$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
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  − 2-s + 4-s − 5-s + 3.21·7-s − 8-s + 10-s − 6.35·11-s + 6.24·13-s − 3.21·14-s + 16-s + 1.21·17-s − 6.95·19-s − 20-s + 6.35·22-s + 4.24·23-s + 25-s − 6.24·26-s + 3.21·28-s − 3.03·29-s + 4.59·31-s − 32-s − 1.21·34-s − 3.21·35-s + 37-s + 6.95·38-s + 40-s + 3.47·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.21·7-s − 0.353·8-s + 0.316·10-s − 1.91·11-s + 1.73·13-s − 0.859·14-s + 0.250·16-s + 0.294·17-s − 1.59·19-s − 0.223·20-s + 1.35·22-s + 0.885·23-s + 0.200·25-s − 1.22·26-s + 0.607·28-s − 0.563·29-s + 0.824·31-s − 0.176·32-s − 0.208·34-s − 0.543·35-s + 0.164·37-s + 1.12·38-s + 0.158·40-s + 0.542·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270159088\)
\(L(\frac12)\) \(\approx\) \(1.270159088\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
37 \( 1 - T \)
good7 \( 1 - 3.21T + 7T^{2} \)
11 \( 1 + 6.35T + 11T^{2} \)
13 \( 1 - 6.24T + 13T^{2} \)
17 \( 1 - 1.21T + 17T^{2} \)
19 \( 1 + 6.95T + 19T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 + 3.03T + 29T^{2} \)
31 \( 1 - 4.59T + 31T^{2} \)
41 \( 1 - 3.47T + 41T^{2} \)
43 \( 1 + 1.47T + 43T^{2} \)
47 \( 1 - 8.01T + 47T^{2} \)
53 \( 1 - 0.773T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 6.28T + 67T^{2} \)
71 \( 1 + 0.882T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 - 2.88T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 - 6.61T + 89T^{2} \)
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

−8.498061073111284012812134858953, −7.998352405335396176359285593262, −7.50287129032826981155794327798, −6.45232362289096563226718120269, −5.63512632162785491799234234037, −4.84453982953496869275825758948, −3.94708077488087859753623687728, −2.82974217841032749273545653126, −1.91446810302686765118371170020, −0.75402223756590700038271023621, 0.75402223756590700038271023621, 1.91446810302686765118371170020, 2.82974217841032749273545653126, 3.94708077488087859753623687728, 4.84453982953496869275825758948, 5.63512632162785491799234234037, 6.45232362289096563226718120269, 7.50287129032826981155794327798, 7.998352405335396176359285593262, 8.498061073111284012812134858953

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