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Properties

Label 2-4608-16.13-c1-0-38
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 − 2.82i·7-s + (3 + 3i)13-s + 4·17-s + (5.65 + 5.65i)19-s + 5.65i·23-s + 3i·25-s + (1 + i)29-s − 2.82·31-s + (−2.82 − 2.82i)35-s + (3 − 3i)37-s + 4i·41-s − 11.3·47-s − 1.00·49-s + (−5 + 5i)53-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s − 1.06i·7-s + (0.832 + 0.832i)13-s + 0.970·17-s + (1.29 + 1.29i)19-s + 1.17i·23-s + 0.600i·25-s + (0.185 + 0.185i)29-s − 0.508·31-s + (−0.478 − 0.478i)35-s + (0.493 − 0.493i)37-s + 0.624i·41-s − 1.65·47-s − 0.142·49-s + (−0.686 + 0.686i)53-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

\(L(1)\) \(\approx\) \(2.283998149\)
\(L(\frac12)\) \(\approx\) \(2.283998149\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (-5.65 - 5.65i)T + 19iT^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + (-1 - i)T + 29iT^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + (8.48 - 8.48i)T - 59iT^{2} \)
61 \( 1 + (1 + i)T + 61iT^{2} \)
67 \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + (11.3 + 11.3i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−8.284612321881998511250895998449, −7.55818439425784279613937405499, −7.10718960083941473934882530702, −5.98744867663181494478613199548, −5.57840344345731255775798798352, −4.61495633903138081475506032159, −3.73759686176021479365313475180, −3.22324899175836172551946218979, −1.53045489176891164453057196729, −1.23150278463987248618962966705, 0.69952977551712132652291930951, 1.99649641577090893055883672527, 2.92203226714161181329521378041, 3.38479871937764520245286781257, 4.80579747490104092016538662075, 5.32909346551212528587484424485, 6.18909911488853481739737855683, 6.57062306305875283728751514209, 7.71856613162628943351003398498, 8.198425794770444238435567134112

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