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Properties

Label 2-4608-16.13-c1-0-51
Degree $2$
Conductor $4608$
Sign $0.923 - 0.382i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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 − i)5-s + 4i·7-s + (4 − 4i)11-s + (3 + 3i)13-s + 6·17-s + (4 + 4i)19-s − 8i·23-s + 3i·25-s + (3 + 3i)29-s − 4·31-s + (4 + 4i)35-s + (1 − i)37-s − 2i·41-s + (4 − 4i)43-s − 8·47-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s + 1.51i·7-s + (1.20 − 1.20i)11-s + (0.832 + 0.832i)13-s + 1.45·17-s + (0.917 + 0.917i)19-s − 1.66i·23-s + 0.600i·25-s + (0.557 + 0.557i)29-s − 0.718·31-s + (0.676 + 0.676i)35-s + (0.164 − 0.164i)37-s − 0.312i·41-s + (0.609 − 0.609i)43-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.695790635\)
\(L(\frac12)\) \(\approx\) \(2.695790635\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + (-4 + 4i)T - 11iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (-4 - 4i)T + 19iT^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + (-3 - 3i)T + 29iT^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 + (-4 + 4i)T - 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (7 - 7i)T - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (3 + 3i)T + 61iT^{2} \)
67 \( 1 + (8 + 8i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (4 + 4i)T + 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.569028267117341858163542905323, −7.84732912415814948894030486778, −6.67591611004168398158490410375, −5.94037993955020654453763340136, −5.71903883616252366042777508005, −4.78481354215264691044535901634, −3.63586968003585771744139013649, −3.08088799824196311276220267706, −1.81042524086582731211086656721, −1.10069760573670121670120322937, 0.955233362683064482232371904822, 1.58221874533219103996538297112, 3.07601062391924803363290300885, 3.64492984390797301479360053883, 4.45528343281635513792574067847, 5.33720277596916778165013425701, 6.23393742597915424383474765373, 6.87265703637919469004909322905, 7.56036711427846394512089693224, 7.947885281363358547833914804977

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