Properties

Label 2-4923-1.1-c1-0-30
Degree $2$
Conductor $4923$
Sign $1$
Analytic cond. $39.3103$
Root an. cond. $6.26979$
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  − 1.15·2-s − 0.659·4-s + 0.421·5-s − 0.645·7-s + 3.07·8-s − 0.487·10-s + 0.640·11-s − 4.07·13-s + 0.747·14-s − 2.24·16-s − 6.47·17-s − 4.15·19-s − 0.277·20-s − 0.741·22-s + 7.48·23-s − 4.82·25-s + 4.71·26-s + 0.425·28-s + 0.298·29-s + 10.2·31-s − 3.55·32-s + 7.49·34-s − 0.271·35-s − 6.30·37-s + 4.80·38-s + 1.29·40-s − 3.79·41-s + ⋯
L(s)  = 1  − 0.818·2-s − 0.329·4-s + 0.188·5-s − 0.243·7-s + 1.08·8-s − 0.154·10-s + 0.192·11-s − 1.12·13-s + 0.199·14-s − 0.561·16-s − 1.57·17-s − 0.952·19-s − 0.0621·20-s − 0.158·22-s + 1.56·23-s − 0.964·25-s + 0.924·26-s + 0.0803·28-s + 0.0553·29-s + 1.83·31-s − 0.628·32-s + 1.28·34-s − 0.0459·35-s − 1.03·37-s + 0.779·38-s + 0.205·40-s − 0.592·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4923\)    =    \(3^{2} \cdot 547\)
Sign: $1$
Analytic conductor: \(39.3103\)
Root analytic conductor: \(6.26979\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4923,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6443991559\)
\(L(\frac12)\) \(\approx\) \(0.6443991559\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
547 \( 1 + T \)
good2 \( 1 + 1.15T + 2T^{2} \)
5 \( 1 - 0.421T + 5T^{2} \)
7 \( 1 + 0.645T + 7T^{2} \)
11 \( 1 - 0.640T + 11T^{2} \)
13 \( 1 + 4.07T + 13T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 + 4.15T + 19T^{2} \)
23 \( 1 - 7.48T + 23T^{2} \)
29 \( 1 - 0.298T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 6.30T + 37T^{2} \)
41 \( 1 + 3.79T + 41T^{2} \)
43 \( 1 - 0.987T + 43T^{2} \)
47 \( 1 + 0.803T + 47T^{2} \)
53 \( 1 - 0.867T + 53T^{2} \)
59 \( 1 + 0.841T + 59T^{2} \)
61 \( 1 + 5.88T + 61T^{2} \)
67 \( 1 - 3.61T + 67T^{2} \)
71 \( 1 - 4.80T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 5.23T + 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 - 9.06T + 89T^{2} \)
97 \( 1 - 5.13T + 97T^{2} \)
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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

−8.336874719831389878971152926403, −7.73684220822253840949347381265, −6.76279752271544558541289206991, −6.47524079422901536295595041670, −5.06402281396394348227254495710, −4.72400628711162006757687814568, −3.81574510819215023068378756180, −2.61399124107256459014430297326, −1.80493294679015455105316572231, −0.49149542253428544341679125791, 0.49149542253428544341679125791, 1.80493294679015455105316572231, 2.61399124107256459014430297326, 3.81574510819215023068378756180, 4.72400628711162006757687814568, 5.06402281396394348227254495710, 6.47524079422901536295595041670, 6.76279752271544558541289206991, 7.73684220822253840949347381265, 8.336874719831389878971152926403

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