Properties

Label 2-48e2-1.1-c1-0-7
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·13-s − 8·17-s − 25-s + 10·29-s + 12·37-s + 8·41-s − 7·49-s − 14·53-s + 12·61-s − 8·65-s + 6·73-s + 16·85-s + 16·89-s + 18·97-s + 2·101-s + 20·109-s + 16·113-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.10·13-s − 1.94·17-s − 1/5·25-s + 1.85·29-s + 1.97·37-s + 1.24·41-s − 49-s − 1.92·53-s + 1.53·61-s − 0.992·65-s + 0.702·73-s + 1.73·85-s + 1.69·89-s + 1.82·97-s + 0.199·101-s + 1.91·109-s + 1.50·113-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: yes
Arithmetic: yes
Character: $\chi_{2304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.367913948\)
\(L(\frac12)\) \(\approx\) \(1.367913948\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−8.861792168523134861661213935674, −8.242790060565164263842729426674, −7.61265699519221135849619148511, −6.51852827413755820930431404947, −6.17025673306416658835216275747, −4.73273267023267405907317724557, −4.25715037631953280136020371932, −3.30329965092489688314729963431, −2.22519740698268929137377439052, −0.75200942853633275817465282286, 0.75200942853633275817465282286, 2.22519740698268929137377439052, 3.30329965092489688314729963431, 4.25715037631953280136020371932, 4.73273267023267405907317724557, 6.17025673306416658835216275747, 6.51852827413755820930431404947, 7.61265699519221135849619148511, 8.242790060565164263842729426674, 8.861792168523134861661213935674

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