Properties

Label 2-4608-16.5-c1-0-19
Degree $2$
Conductor $4608$
Sign $-0.707 - 0.707i$
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  + (2.41 + 2.41i)5-s + 1.53i·7-s + (3.37 + 3.37i)11-s + (−0.414 + 0.414i)13-s − 2.82·17-s + (0.317 − 0.317i)19-s + 5.86i·23-s + 6.65i·25-s + (−3.24 + 3.24i)29-s − 7.39·31-s + (−3.69 + 3.69i)35-s + (3.58 + 3.58i)37-s − 4i·41-s + (−1.84 − 1.84i)43-s − 7.39·47-s + ⋯
L(s)  = 1  + (1.07 + 1.07i)5-s + 0.578i·7-s + (1.01 + 1.01i)11-s + (−0.114 + 0.114i)13-s − 0.685·17-s + (0.0727 − 0.0727i)19-s + 1.22i·23-s + 1.33i·25-s + (−0.602 + 0.602i)29-s − 1.32·31-s + (−0.624 + 0.624i)35-s + (0.589 + 0.589i)37-s − 0.624i·41-s + (−0.281 − 0.281i)43-s − 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-0.707 - 0.707i$
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: \((2,\ 4608,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.084058238\)
\(L(\frac12)\) \(\approx\) \(2.084058238\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 2.41i)T + 5iT^{2} \)
7 \( 1 - 1.53iT - 7T^{2} \)
11 \( 1 + (-3.37 - 3.37i)T + 11iT^{2} \)
13 \( 1 + (0.414 - 0.414i)T - 13iT^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (-0.317 + 0.317i)T - 19iT^{2} \)
23 \( 1 - 5.86iT - 23T^{2} \)
29 \( 1 + (3.24 - 3.24i)T - 29iT^{2} \)
31 \( 1 + 7.39T + 31T^{2} \)
37 \( 1 + (-3.58 - 3.58i)T + 37iT^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + (1.84 + 1.84i)T + 43iT^{2} \)
47 \( 1 + 7.39T + 47T^{2} \)
53 \( 1 + (5.24 + 5.24i)T + 53iT^{2} \)
59 \( 1 + (-1.84 - 1.84i)T + 59iT^{2} \)
61 \( 1 + (-9.24 + 9.24i)T - 61iT^{2} \)
67 \( 1 + (7.07 - 7.07i)T - 67iT^{2} \)
71 \( 1 - 11.9iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 - 6.12T + 79T^{2} \)
83 \( 1 + (-2.48 + 2.48i)T - 83iT^{2} \)
89 \( 1 - 0.828iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.853883258090056620556514126363, −7.66919808743913834659417920329, −6.93012352123695659932498373152, −6.56621753969940048806758618742, −5.70823740106185254324628375391, −5.08466765559603804156710951399, −3.97016116717315529250795474472, −3.15851811930828321152686284016, −2.10405051288651657384971029825, −1.69905062935868416396924989287, 0.54063107941873493671479739961, 1.43709692688999567739477180668, 2.36113298531816451154477593836, 3.58630434061054085061895994858, 4.35740021872774184338001790055, 5.08514139021306583495520215863, 5.98612621138381141578649470450, 6.34165193119387168058702256686, 7.30756883354144035550004344083, 8.235539730818148961542064969444

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