Properties

Label 2-48e2-1.1-c1-0-16
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  + 3.46·7-s + 6.92·13-s + 8·19-s − 5·25-s − 10.3·31-s − 6.92·37-s + 8·43-s + 4.99·49-s − 6.92·61-s + 16·67-s + 10·73-s + 17.3·79-s + 23.9·91-s − 14·97-s − 3.46·103-s − 20.7·109-s + ⋯
L(s)  = 1  + 1.30·7-s + 1.92·13-s + 1.83·19-s − 25-s − 1.86·31-s − 1.13·37-s + 1.21·43-s + 0.714·49-s − 0.887·61-s + 1.95·67-s + 1.17·73-s + 1.94·79-s + 2.51·91-s − 1.42·97-s − 0.341·103-s − 1.99·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.386914906\)
\(L(\frac12)\) \(\approx\) \(2.386914906\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.92T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 - 16T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.992519411292991924842245807614, −8.119739549222488436583267113574, −7.68001558183480047677398162889, −6.72350388056099916704719561778, −5.62322477351063974061968965290, −5.26451074401308516574797775769, −4.02597593827844372717627042427, −3.41532837479100420248248570026, −1.93459654960902440686875369465, −1.11584692571036805227010716670, 1.11584692571036805227010716670, 1.93459654960902440686875369465, 3.41532837479100420248248570026, 4.02597593827844372717627042427, 5.26451074401308516574797775769, 5.62322477351063974061968965290, 6.72350388056099916704719561778, 7.68001558183480047677398162889, 8.119739549222488436583267113574, 8.992519411292991924842245807614

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