Properties

Label 2-5100-1.1-c1-0-20
Degree $2$
Conductor $5100$
Sign $1$
Analytic cond. $40.7237$
Root an. cond. $6.38151$
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  + 3-s + 1.12·7-s + 9-s − 0.781·11-s + 6.90·13-s − 17-s − 4.99·19-s + 1.12·21-s + 2.69·23-s + 27-s − 4.48·29-s + 5.36·31-s − 0.781·33-s − 1.87·37-s + 6.90·39-s + 12.1·41-s − 4.10·43-s + 12.1·47-s − 5.73·49-s − 51-s + 7.12·53-s − 4.99·57-s + 8.61·59-s − 13.0·61-s + 1.12·63-s − 8.61·67-s + 2.69·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.424·7-s + 0.333·9-s − 0.235·11-s + 1.91·13-s − 0.242·17-s − 1.14·19-s + 0.245·21-s + 0.562·23-s + 0.192·27-s − 0.833·29-s + 0.963·31-s − 0.136·33-s − 0.308·37-s + 1.10·39-s + 1.90·41-s − 0.625·43-s + 1.76·47-s − 0.819·49-s − 0.140·51-s + 0.978·53-s − 0.660·57-s + 1.12·59-s − 1.66·61-s + 0.141·63-s − 1.05·67-s + 0.324·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(40.7237\)
Root analytic conductor: \(6.38151\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.853658878\)
\(L(\frac12)\) \(\approx\) \(2.853658878\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
17 \( 1 + T \)
good7 \( 1 - 1.12T + 7T^{2} \)
11 \( 1 + 0.781T + 11T^{2} \)
13 \( 1 - 6.90T + 13T^{2} \)
19 \( 1 + 4.99T + 19T^{2} \)
23 \( 1 - 2.69T + 23T^{2} \)
29 \( 1 + 4.48T + 29T^{2} \)
31 \( 1 - 5.36T + 31T^{2} \)
37 \( 1 + 1.87T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + 4.10T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 - 8.61T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 8.61T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 7.29T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
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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

−8.264630366534032086901416333290, −7.70944008971150924693770881347, −6.75137904908260658846669924449, −6.14262763647478122432066712986, −5.34346001837206476991025358882, −4.29255802487438778695944304928, −3.83090617083835083048326069635, −2.82432007890946143206135795104, −1.93284826448060110361544688751, −0.937155415598296276944763775701, 0.937155415598296276944763775701, 1.93284826448060110361544688751, 2.82432007890946143206135795104, 3.83090617083835083048326069635, 4.29255802487438778695944304928, 5.34346001837206476991025358882, 6.14262763647478122432066712986, 6.75137904908260658846669924449, 7.70944008971150924693770881347, 8.264630366534032086901416333290

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