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Properties

Label 2-48e2-8.5-c1-0-21
Degree $2$
Conductor $2304$
Sign $0.707 + 0.707i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·5-s − 6i·13-s − 2·17-s + 25-s − 10i·29-s − 2i·37-s + 10·41-s − 7·49-s − 14i·53-s + 10i·61-s + 12·65-s + 6·73-s − 4i·85-s + 10·89-s + 18·97-s + ⋯
L(s)  = 1  + 0.894i·5-s − 1.66i·13-s − 0.485·17-s + 0.200·25-s − 1.85i·29-s − 0.328i·37-s + 1.56·41-s − 49-s − 1.92i·53-s + 1.28i·61-s + 1.48·65-s + 0.702·73-s − 0.433i·85-s + 1.05·89-s + 1.82·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513845634\)
\(L(\frac12)\) \(\approx\) \(1.513845634\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 10iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−8.870122150054973899186680766434, −7.972463656754077885956871992656, −7.49479572250425860158431939397, −6.50591638218324463835086773241, −5.90392082831216485065391759704, −4.98585295206129508671893881331, −3.91441138178663553219635955667, −3.01564589045089434139769274086, −2.26426503855373294487258093476, −0.57734291948025958873586599270, 1.15331537388626978908776570759, 2.13705874843897989731914521872, 3.42993865374268533565915872864, 4.50629054228408865342434215136, 4.88432803683481846377458524126, 6.02669960499771947558155484729, 6.77718234846298638342818006281, 7.54123027625394416507868563524, 8.561557042364399261531793893122, 9.078425506673035674711490000200

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