Properties

Label 2-48e2-1.1-c1-0-19
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  + 3.46·5-s + 3.46·7-s − 6·17-s + 4·19-s + 6.92·23-s + 6.99·25-s − 3.46·29-s + 3.46·31-s + 11.9·35-s + 6.92·37-s − 6·41-s + 4·43-s − 6.92·47-s + 4.99·49-s − 3.46·53-s + 12·59-s − 6.92·61-s − 4·67-s − 6.92·71-s − 2·73-s − 10.3·79-s − 20.7·85-s + 6·89-s + 13.8·95-s − 2·97-s − 3.46·101-s − 17.3·103-s + ⋯
L(s)  = 1  + 1.54·5-s + 1.30·7-s − 1.45·17-s + 0.917·19-s + 1.44·23-s + 1.39·25-s − 0.643·29-s + 0.622·31-s + 2.02·35-s + 1.13·37-s − 0.937·41-s + 0.609·43-s − 1.01·47-s + 0.714·49-s − 0.475·53-s + 1.56·59-s − 0.887·61-s − 0.488·67-s − 0.822·71-s − 0.234·73-s − 1.16·79-s − 2.25·85-s + 0.635·89-s + 1.42·95-s − 0.203·97-s − 0.344·101-s − 1.70·103-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

\(L(1)\) \(\approx\) \(2.887685050\)
\(L(\frac12)\) \(\approx\) \(2.887685050\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 3.46T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.051958995200110935297158929877, −8.393135543397082230969700621486, −7.40784544888046377444052101226, −6.63533842994695874337372444054, −5.78738706852668711508938914379, −5.05640537320441894785617631505, −4.47073029822226100270455099896, −2.95041560502240512540940991021, −2.04697187072846738510564596377, −1.24474527390506146238494436048, 1.24474527390506146238494436048, 2.04697187072846738510564596377, 2.95041560502240512540940991021, 4.47073029822226100270455099896, 5.05640537320441894785617631505, 5.78738706852668711508938914379, 6.63533842994695874337372444054, 7.40784544888046377444052101226, 8.393135543397082230969700621486, 9.051958995200110935297158929877

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