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Properties

Label 2-48e2-16.5-c1-0-31
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 + 2i)5-s − 4.24i·7-s + (−2.82 − 2.82i)11-s + (−3 + 3i)13-s + 6·17-s + (−1.41 + 1.41i)19-s − 2.82i·23-s + 3i·25-s + (4 − 4i)29-s + 4.24·31-s + (8.48 − 8.48i)35-s + (−3 − 3i)37-s − 10i·41-s + (−4.24 − 4.24i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (0.894 + 0.894i)5-s − 1.60i·7-s + (−0.852 − 0.852i)11-s + (−0.832 + 0.832i)13-s + 1.45·17-s + (−0.324 + 0.324i)19-s − 0.589i·23-s + 0.600i·25-s + (0.742 − 0.742i)29-s + 0.762·31-s + (1.43 − 1.43i)35-s + (−0.493 − 0.493i)37-s − 1.56i·41-s + (−0.646 − 0.646i)43-s + 0.412·47-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

\(L(1)\) \(\approx\) \(1.760393218\)
\(L(\frac12)\) \(\approx\) \(1.760393218\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 2i)T + 5iT^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + (2.82 + 2.82i)T + 11iT^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (1.41 - 1.41i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-4 + 4i)T - 29iT^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + (4.24 + 4.24i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-4 - 4i)T + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (3 - 3i)T - 61iT^{2} \)
67 \( 1 + (-2.82 + 2.82i)T - 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + (-11.3 + 11.3i)T - 83iT^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−8.908074962374207124963676068820, −7.85479426904919696777756139442, −7.33173702209483314372019319424, −6.56294294086149800767323602736, −5.85994842223965512231748820005, −4.88400112366028489997712661517, −3.89394196296759905334504483012, −3.01101206757147578648446175860, −2.04690827191112336947048231876, −0.61146414010691694521683504386, 1.32938855879292541447927399206, 2.40815721792198447247373615429, 3.04684825007837185765112680077, 4.80397716659773166444022497642, 5.24520231835866757431053605976, 5.69575739591122151734005246437, 6.72932831225017469164862117411, 7.896470253653029720015415973446, 8.339777616568070192255055859265, 9.248309714360219442859240663211

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