Properties

Label 2-3312-1.1-c1-0-15
Degree $2$
Conductor $3312$
Sign $1$
Analytic cond. $26.4464$
Root an. cond. $5.14261$
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  − 1.64·5-s + 1.01·7-s + 5.21·11-s + 2.55·13-s + 3.56·17-s − 3.09·19-s + 23-s − 2.29·25-s + 1.44·29-s − 1.44·31-s − 1.66·35-s + 6.10·37-s + 4.73·41-s − 11.3·43-s − 0.0221·47-s − 5.97·49-s + 7.66·53-s − 8.57·55-s + 3.28·59-s + 7.21·61-s − 4.20·65-s − 11.4·67-s − 0.736·71-s + 11.6·73-s + 5.26·77-s + 2.12·79-s + 9.21·83-s + ⋯
L(s)  = 1  − 0.736·5-s + 0.382·7-s + 1.57·11-s + 0.708·13-s + 0.864·17-s − 0.709·19-s + 0.208·23-s − 0.458·25-s + 0.268·29-s − 0.259·31-s − 0.281·35-s + 1.00·37-s + 0.739·41-s − 1.72·43-s − 0.00323·47-s − 0.853·49-s + 1.05·53-s − 1.15·55-s + 0.427·59-s + 0.923·61-s − 0.521·65-s − 1.40·67-s − 0.0874·71-s + 1.36·73-s + 0.600·77-s + 0.238·79-s + 1.01·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(26.4464\)
Root analytic conductor: \(5.14261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.948097043\)
\(L(\frac12)\) \(\approx\) \(1.948097043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 1.64T + 5T^{2} \)
7 \( 1 - 1.01T + 7T^{2} \)
11 \( 1 - 5.21T + 11T^{2} \)
13 \( 1 - 2.55T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
19 \( 1 + 3.09T + 19T^{2} \)
29 \( 1 - 1.44T + 29T^{2} \)
31 \( 1 + 1.44T + 31T^{2} \)
37 \( 1 - 6.10T + 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 0.0221T + 47T^{2} \)
53 \( 1 - 7.66T + 53T^{2} \)
59 \( 1 - 3.28T + 59T^{2} \)
61 \( 1 - 7.21T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 0.736T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 2.12T + 79T^{2} \)
83 \( 1 - 9.21T + 83T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + 7.31T + 97T^{2} \)
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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

−8.490892432170127232303134566773, −8.035661720421078777649551532649, −7.13137043739072417444648524538, −6.43378470333865677237585725110, −5.69754992257747941192472380377, −4.59664950795368886719792791491, −3.93095555862608866189639992860, −3.29706013685581951883348985007, −1.87858449954833478737123682163, −0.880996477802219952126540561766, 0.880996477802219952126540561766, 1.87858449954833478737123682163, 3.29706013685581951883348985007, 3.93095555862608866189639992860, 4.59664950795368886719792791491, 5.69754992257747941192472380377, 6.43378470333865677237585725110, 7.13137043739072417444648524538, 8.035661720421078777649551532649, 8.490892432170127232303134566773

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