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Properties

Label 2-2664-37.26-c1-0-32
Degree $2$
Conductor $2664$
Sign $0.729 + 0.683i$
Analytic cond. $21.2721$
Root an. cond. $4.61217$
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 − 3.46i)5-s + (−0.5 + 0.866i)7-s + 2·11-s + (−1.5 + 2.59i)13-s + (4 + 6.92i)17-s + (4 − 6.92i)19-s + (−5.49 − 9.52i)25-s + 10·29-s − 7·31-s + (1.99 + 3.46i)35-s + (5 + 3.46i)37-s + (−1 + 1.73i)41-s + 5·43-s + 6·47-s + (3 + 5.19i)49-s + ⋯
L(s)  = 1  + (0.894 − 1.54i)5-s + (−0.188 + 0.327i)7-s + 0.603·11-s + (−0.416 + 0.720i)13-s + (0.970 + 1.68i)17-s + (0.917 − 1.58i)19-s + (−1.09 − 1.90i)25-s + 1.85·29-s − 1.25·31-s + (0.338 + 0.585i)35-s + (0.821 + 0.569i)37-s + (−0.156 + 0.270i)41-s + 0.762·43-s + 0.875·47-s + (0.428 + 0.742i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.729 + 0.683i$
Analytic conductor: \(21.2721\)
Root analytic conductor: \(4.61217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :1/2),\ 0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.337576051\)
\(L(\frac12)\) \(\approx\) \(2.337576051\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + (-5 - 3.46i)T \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5 + 8.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−8.930267028895457925206521863206, −8.240222300797045691805835038423, −7.24463525534480450554842888535, −6.21488307505504371887496639473, −5.71890343084899947023491434100, −4.80472964170495918478561105205, −4.25066591856404981381495416740, −2.91872214889269100483356165941, −1.75441238870526024657891730155, −0.958811296649683288568431417256, 1.10536310643562621886973128308, 2.49516652736310633098288865380, 3.09512398078593486238080053431, 3.93087850472425164840570110962, 5.40235722226429424118233048168, 5.77703665255426160885376950029, 6.79493446568862523005301218183, 7.28531753485708225392702017273, 7.921425022210030726747763241306, 9.255865215419228105865534724722

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