Properties

Label 2-48e2-16.5-c1-0-6
Degree $2$
Conductor $2304$
Sign $-0.608 - 0.793i$
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 + 3.86i·7-s + (3.34 + 3.34i)11-s + (0.267 − 0.267i)13-s − 3.46·17-s + (3.34 − 3.34i)19-s + 1.79i·23-s + 0.999i·25-s + (−1.73 + 1.73i)29-s − 5.65·31-s + (6.69 − 6.69i)35-s + (−3.73 − 3.73i)37-s + 6.92i·41-s + (1.55 + 1.55i)43-s − 9.79·47-s + ⋯
L(s)  = 1  + (−0.774 − 0.774i)5-s + 1.46i·7-s + (1.00 + 1.00i)11-s + (0.0743 − 0.0743i)13-s − 0.840·17-s + (0.767 − 0.767i)19-s + 0.373i·23-s + 0.199i·25-s + (−0.321 + 0.321i)29-s − 1.01·31-s + (1.13 − 1.13i)35-s + (−0.613 − 0.613i)37-s + 1.08i·41-s + (0.236 + 0.236i)43-s − 1.42·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.608 - 0.793i$
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: \((2,\ 2304,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8520354447\)
\(L(\frac12)\) \(\approx\) \(0.8520354447\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.86iT - 7T^{2} \)
11 \( 1 + (-3.34 - 3.34i)T + 11iT^{2} \)
13 \( 1 + (-0.267 + 0.267i)T - 13iT^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + (-3.34 + 3.34i)T - 19iT^{2} \)
23 \( 1 - 1.79iT - 23T^{2} \)
29 \( 1 + (1.73 - 1.73i)T - 29iT^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (3.73 + 3.73i)T + 37iT^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + (-1.55 - 1.55i)T + 43iT^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + (-7.73 - 7.73i)T + 53iT^{2} \)
59 \( 1 + (5.13 + 5.13i)T + 59iT^{2} \)
61 \( 1 + (3.73 - 3.73i)T - 61iT^{2} \)
67 \( 1 + (4.38 - 4.38i)T - 67iT^{2} \)
71 \( 1 - 1.79iT - 71T^{2} \)
73 \( 1 - 2.53iT - 73T^{2} \)
79 \( 1 + 4.14T + 79T^{2} \)
83 \( 1 + (-1.55 + 1.55i)T - 83iT^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
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

−9.074775568322602959828118460424, −8.774778503454303901402852806171, −7.78087017724230490009471344856, −7.02700707623955562212219447897, −6.13380722174991822195768052542, −5.18160671802197310193005356542, −4.58345855508541843438815733609, −3.65233398849174345242219671563, −2.48380012711981374884420161198, −1.42996045479521252568934637351, 0.30203039268803766330076229697, 1.60418843873123878521010013828, 3.28474039572040528350506426877, 3.69134641585397494895951910255, 4.43115590580602700147003828512, 5.69985764377751220805288758758, 6.67907121157428965756044651896, 7.10992225502071971226914065632, 7.84300370186465626397074490758, 8.649105980605030365675377593145

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