Properties

Label 2-4608-16.5-c1-0-28
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 + (−5 + 5i)13-s − 8·17-s − 3i·25-s + (3 − 3i)29-s + (7 + 7i)37-s − 8i·41-s + 7·49-s + (9 + 9i)53-s + (11 − 11i)61-s + 10·65-s − 6i·73-s + (8 + 8i)85-s + 10i·89-s − 8·97-s + ⋯
L(s)  = 1  + (−0.447 − 0.447i)5-s + (−1.38 + 1.38i)13-s − 1.94·17-s − 0.600i·25-s + (0.557 − 0.557i)29-s + (1.15 + 1.15i)37-s − 1.24i·41-s + 49-s + (1.23 + 1.23i)53-s + (1.40 − 1.40i)61-s + 1.24·65-s − 0.702i·73-s + (0.867 + 0.867i)85-s + 1.05i·89-s − 0.812·97-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: $\chi_{4608} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.167334562\)
\(L(\frac12)\) \(\approx\) \(1.167334562\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 + (5 - 5i)T - 13iT^{2} \)
17 \( 1 + 8T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-7 - 7i)T + 37iT^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (-11 + 11i)T - 61iT^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
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

−8.383029175909405037583411973201, −7.48857037009772824668698172985, −6.83734430494136487379030723407, −6.27620456058626321556375541378, −5.11104595264915916438082198079, −4.41248316446448755597460755748, −4.09309636903869768195423940014, −2.59231654955138619059334936847, −2.05254973620759833811881740968, −0.51975964016386229079100706753, 0.63392393671241026761669066715, 2.26802468492041229284588118948, 2.81850783061196489178662367365, 3.83672727783151552714460959956, 4.65457869179857705037560389834, 5.36295181086017796248768573416, 6.23343047959648399832907181062, 7.18257422426829392622792156329, 7.40083421460210234008308722372, 8.385479279033368861609707424836

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