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Properties

Label 2-6624-552.413-c1-0-79
Degree $2$
Conductor $6624$
Sign $-0.577 + 0.816i$
Analytic cond. $52.8929$
Root an. cond. $7.27275$
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  − 3.84i·5-s − 3.84i·11-s + 7.56·19-s − 4.79i·23-s − 9.78·25-s + 10.7·31-s + 2.12·37-s + 9.59i·41-s + 8.74·43-s − 11.0i·47-s + 7·49-s − 14.5i·53-s − 14.7·55-s + 14.1·61-s + 2.12·67-s + ⋯
L(s)  = 1  − 1.71i·5-s − 1.15i·11-s + 1.73·19-s − 0.999i·23-s − 1.95·25-s + 1.93·31-s + 0.349·37-s + 1.49i·41-s + 1.33·43-s − 1.60i·47-s + 49-s − 1.99i·53-s − 1.99·55-s + 1.81·61-s + 0.259·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6624\)    =    \(2^{5} \cdot 3^{2} \cdot 23\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(52.8929\)
Root analytic conductor: \(7.27275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6624} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6624,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.181745908\)
\(L(\frac12)\) \(\approx\) \(2.181745908\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 + 4.79iT \)
good5 \( 1 + 3.84iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 3.84iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.56T + 19T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 - 2.12T + 37T^{2} \)
41 \( 1 - 9.59iT - 41T^{2} \)
43 \( 1 - 8.74T + 43T^{2} \)
47 \( 1 + 11.0iT - 47T^{2} \)
53 \( 1 + 14.5iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 2.12T + 67T^{2} \)
71 \( 1 + 2.52iT - 71T^{2} \)
73 \( 1 + 16.7T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 8.52iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.108663157553037020385354175693, −7.09659063359331676319308413983, −6.16661072289369514782898138017, −5.51832890798406076203519452790, −4.94922432390658933725429378551, −4.25278464712142102488078491710, −3.38043627754276415359712061918, −2.41704601524372456331671402642, −1.06673186761583962301988653963, −0.69030068995264827348152382636, 1.21136339826902263324075316240, 2.44817722202257892117400096961, 2.88607747537262236511991851342, 3.79064429254234172052028904109, 4.54788290044422167783593090574, 5.61158246320247101493473925242, 6.13191823721685849646146820163, 7.15474781604012902921347953787, 7.28927785544833105873687384590, 7.897108454242971279643513717984

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