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Properties

Label 2-48e2-16.13-c1-0-24
Degree $2$
Conductor $2304$
Sign $0.991 + 0.130i$
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  + (1.73 − 1.73i)5-s + 1.03i·7-s + (0.896 − 0.896i)11-s + (3.73 + 3.73i)13-s + 3.46·17-s + (0.896 + 0.896i)19-s − 6.69i·23-s − 0.999i·25-s + (1.73 + 1.73i)29-s − 5.65·31-s + (1.79 + 1.79i)35-s + (−0.267 + 0.267i)37-s + 6.92i·41-s + (−5.79 + 5.79i)43-s + 9.79·47-s + ⋯
L(s)  = 1  + (0.774 − 0.774i)5-s + 0.391i·7-s + (0.270 − 0.270i)11-s + (1.03 + 1.03i)13-s + 0.840·17-s + (0.205 + 0.205i)19-s − 1.39i·23-s − 0.199i·25-s + (0.321 + 0.321i)29-s − 1.01·31-s + (0.303 + 0.303i)35-s + (−0.0440 + 0.0440i)37-s + 1.08i·41-s + (−0.883 + 0.883i)43-s + 1.42·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.315357743\)
\(L(\frac12)\) \(\approx\) \(2.315357743\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 1.73i)T - 5iT^{2} \)
7 \( 1 - 1.03iT - 7T^{2} \)
11 \( 1 + (-0.896 + 0.896i)T - 11iT^{2} \)
13 \( 1 + (-3.73 - 3.73i)T + 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (-0.896 - 0.896i)T + 19iT^{2} \)
23 \( 1 + 6.69iT - 23T^{2} \)
29 \( 1 + (-1.73 - 1.73i)T + 29iT^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (0.267 - 0.267i)T - 37iT^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + (5.79 - 5.79i)T - 43iT^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (-4.26 + 4.26i)T - 53iT^{2} \)
59 \( 1 + (7.58 - 7.58i)T - 59iT^{2} \)
61 \( 1 + (0.267 + 0.267i)T + 61iT^{2} \)
67 \( 1 + (-2.96 - 2.96i)T + 67iT^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 + 9.46iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (5.79 + 5.79i)T + 83iT^{2} \)
89 \( 1 + 9.46iT - 89T^{2} \)
97 \( 1 + 3.46T + 97T^{2} \)
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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

−8.898248566011309195487088649446, −8.549569062387550296059352565758, −7.47967954271027754055594142340, −6.41795632874684820568220674651, −5.92471593432632995252108488841, −5.08344906367158712949443966705, −4.22164124306369339138434846787, −3.19510477554670239806469958808, −1.94767755341236258395659313190, −1.08817257825420553249453433477, 1.03271420344410509539693327117, 2.20179803789562043315753867425, 3.31176781433289262849353508162, 3.91190137638746614709552466379, 5.41878490054751956133662048672, 5.74059026742407324108550914989, 6.75350550575802161278810094107, 7.39387693164359680631223810186, 8.179347325523314951615630547734, 9.156874224825961610470704433814

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