Properties

Label 2-3087-1.1-c1-0-82
Degree $2$
Conductor $3087$
Sign $-1$
Analytic cond. $24.6498$
Root an. cond. $4.96485$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.69·2-s + 5.24·4-s − 8.74·8-s + 5.89·11-s + 13.0·16-s − 15.8·22-s − 3.37·23-s − 5·25-s − 9.87·29-s − 17.6·32-s + 0.814·37-s − 3.34·43-s + 30.9·44-s + 9.09·46-s + 13.4·50-s − 2.03·53-s + 26.5·58-s + 21.3·64-s − 16.3·67-s − 14.1·71-s − 2.19·74-s − 7.42·79-s + 9.00·86-s − 51.5·88-s − 17.7·92-s − 26.2·100-s + 5.48·106-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.62·4-s − 3.09·8-s + 1.77·11-s + 3.25·16-s − 3.38·22-s − 0.704·23-s − 25-s − 1.83·29-s − 3.11·32-s + 0.133·37-s − 0.510·43-s + 4.66·44-s + 1.34·46-s + 1.90·50-s − 0.280·53-s + 3.48·58-s + 2.66·64-s − 1.99·67-s − 1.67·71-s − 0.254·74-s − 0.835·79-s + 0.970·86-s − 5.49·88-s − 1.84·92-s − 2.62·100-s + 0.533·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3087\)    =    \(3^{2} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(24.6498\)
Root analytic conductor: \(4.96485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3087,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 5.89T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 3.37T + 23T^{2} \)
29 \( 1 + 9.87T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.814T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 3.34T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 2.03T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 16.3T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 7.42T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.532049309798470089052488358083, −7.64021378728423330942212927243, −7.16626128814891185361072003304, −6.27998907152896842068997125492, −5.80784063618549732365084195631, −4.17885337766834545879160965536, −3.28432283095661615451271831493, −1.99365323003417775263957010818, −1.37790193576724217795863225762, 0, 1.37790193576724217795863225762, 1.99365323003417775263957010818, 3.28432283095661615451271831493, 4.17885337766834545879160965536, 5.80784063618549732365084195631, 6.27998907152896842068997125492, 7.16626128814891185361072003304, 7.64021378728423330942212927243, 8.532049309798470089052488358083

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