Properties

Label 2-96e2-1.1-c1-0-18
Degree $2$
Conductor $9216$
Sign $1$
Analytic cond. $73.5901$
Root an. cond. $8.57846$
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 − 4·7-s − 5.65·11-s + 4.24·13-s − 6·17-s + 5.65·19-s − 8·23-s − 2.99·25-s − 4.24·29-s − 4·31-s − 5.65·35-s + 1.41·37-s − 2·41-s − 5.65·43-s + 8·47-s + 9·49-s + 9.89·53-s − 8.00·55-s + 4.24·61-s + 6·65-s + 11.3·67-s + 10·73-s + 22.6·77-s − 12·79-s − 5.65·83-s − 8.48·85-s + 16·89-s + ⋯
L(s)  = 1  + 0.632·5-s − 1.51·7-s − 1.70·11-s + 1.17·13-s − 1.45·17-s + 1.29·19-s − 1.66·23-s − 0.599·25-s − 0.787·29-s − 0.718·31-s − 0.956·35-s + 0.232·37-s − 0.312·41-s − 0.862·43-s + 1.16·47-s + 1.28·49-s + 1.35·53-s − 1.07·55-s + 0.543·61-s + 0.744·65-s + 1.38·67-s + 1.17·73-s + 2.57·77-s − 1.35·79-s − 0.620·83-s − 0.920·85-s + 1.69·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(73.5901\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9216,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9962343167\)
\(L(\frac12)\) \(\approx\) \(0.9962343167\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 1.41T + 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 1.41T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 9.89T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 4.24T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−7.66330791894148370073214364521, −6.98763763386974586798955879502, −6.22138687512023386680462486194, −5.75275284959785702500702929207, −5.21552013547093499258210996749, −3.99684416024933049624871436888, −3.48164130411424294812662879242, −2.54922596404450608833455710750, −1.95971681267844235676834479855, −0.44847725867758537800451098283, 0.44847725867758537800451098283, 1.95971681267844235676834479855, 2.54922596404450608833455710750, 3.48164130411424294812662879242, 3.99684416024933049624871436888, 5.21552013547093499258210996749, 5.75275284959785702500702929207, 6.22138687512023386680462486194, 6.98763763386974586798955879502, 7.66330791894148370073214364521

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