Properties

Label 2-3528-1.1-c1-0-16
Degree $2$
Conductor $3528$
Sign $1$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
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  + 1.41·5-s + 1.41·13-s − 1.41·17-s + 4·23-s − 2.99·25-s + 6·29-s + 5.65·31-s − 4·37-s − 7.07·41-s + 4·43-s + 11.3·47-s + 4·53-s − 5.65·59-s + 4.24·61-s + 2.00·65-s + 4·67-s + 12·71-s − 7.07·73-s + 8·79-s − 5.65·83-s − 2.00·85-s + 12.7·89-s + 4.24·97-s − 9.89·101-s − 11.3·103-s + 16·107-s − 4·109-s + ⋯
L(s)  = 1  + 0.632·5-s + 0.392·13-s − 0.342·17-s + 0.834·23-s − 0.599·25-s + 1.11·29-s + 1.01·31-s − 0.657·37-s − 1.10·41-s + 0.609·43-s + 1.65·47-s + 0.549·53-s − 0.736·59-s + 0.543·61-s + 0.248·65-s + 0.488·67-s + 1.42·71-s − 0.827·73-s + 0.900·79-s − 0.620·83-s − 0.216·85-s + 1.34·89-s + 0.430·97-s − 0.985·101-s − 1.11·103-s + 1.54·107-s − 0.383·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.203680966\)
\(L(\frac12)\) \(\approx\) \(2.203680966\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.41T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 - 4.24T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 7.07T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−8.657284465829937159154535871358, −7.898343611211250987278091707209, −6.95208894495956377479898243742, −6.37388625516611858227023080884, −5.58192272729343979893803934832, −4.82833326096884863195989945874, −3.92842572480601102662905257371, −2.92312634098921623400413298936, −2.04358096292737612482560710779, −0.898270004494159331583658136594, 0.898270004494159331583658136594, 2.04358096292737612482560710779, 2.92312634098921623400413298936, 3.92842572480601102662905257371, 4.82833326096884863195989945874, 5.58192272729343979893803934832, 6.37388625516611858227023080884, 6.95208894495956377479898243742, 7.898343611211250987278091707209, 8.657284465829937159154535871358

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