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Properties

Label 2-48e2-16.5-c1-0-20
Degree $2$
Conductor $2304$
Sign $0.382 - 0.923i$
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  + (2.44 + 2.44i)5-s + 1.41i·7-s + (3.46 + 3.46i)11-s + (−1 + i)13-s + 4.89·17-s + (4.24 − 4.24i)19-s − 6.92i·23-s + 6.99i·25-s + (2.44 − 2.44i)29-s − 1.41·31-s + (−3.46 + 3.46i)35-s + (7 + 7i)37-s + 4.89i·41-s + (−4.24 − 4.24i)43-s − 6.92·47-s + ⋯
L(s)  = 1  + (1.09 + 1.09i)5-s + 0.534i·7-s + (1.04 + 1.04i)11-s + (−0.277 + 0.277i)13-s + 1.18·17-s + (0.973 − 0.973i)19-s − 1.44i·23-s + 1.39i·25-s + (0.454 − 0.454i)29-s − 0.254·31-s + (−0.585 + 0.585i)35-s + (1.15 + 1.15i)37-s + 0.765i·41-s + (−0.646 − 0.646i)43-s − 1.01·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.514259014\)
\(L(\frac12)\) \(\approx\) \(2.514259014\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 - 2.44i)T + 5iT^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 + (-3.46 - 3.46i)T + 11iT^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 + (-4.24 + 4.24i)T - 19iT^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + (-2.44 + 2.44i)T - 29iT^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + (-7 - 7i)T + 37iT^{2} \)
41 \( 1 - 4.89iT - 41T^{2} \)
43 \( 1 + (4.24 + 4.24i)T + 43iT^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + (2.44 + 2.44i)T + 53iT^{2} \)
59 \( 1 + (6.92 + 6.92i)T + 59iT^{2} \)
61 \( 1 + (5 - 5i)T - 61iT^{2} \)
67 \( 1 + (-5.65 + 5.65i)T - 67iT^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + (-10.3 + 10.3i)T - 83iT^{2} \)
89 \( 1 + 9.79iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.487046379169819664807538692188, −8.453564843979592734589419622563, −7.40135980712424972672807328673, −6.68973529506364756328895165344, −6.24355040894732990825513785778, −5.26028852004658240374846416466, −4.41657364598535375115895386963, −3.09613472086421004419943726993, −2.46460684292076548120397777403, −1.41265193476939621273408601493, 1.01330456036734425147560323510, 1.55016276352500378858196602241, 3.16635070409653421668687653711, 3.91747163867007589535351529227, 5.12061266887827726795817221233, 5.66876345632782040261208205527, 6.26946445336178959833555268548, 7.49362936295663227407650833190, 8.066990510012932111398128887740, 9.128757798541103631146528736544

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