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Properties

Label 2-2664-296.211-c0-0-3
Degree $2$
Conductor $2664$
Sign $-0.803 + 0.595i$
Analytic cond. $1.32950$
Root an. cond. $1.15304$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−1.22 − 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.999 − i)10-s − 1.41·11-s + (−0.866 − 0.5i)13-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (−1 + 1.73i)19-s + (−0.707 − 1.22i)20-s + (−1.36 − 0.366i)22-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−1.22 − 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.999 − i)10-s − 1.41·11-s + (−0.866 − 0.5i)13-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (−1 + 1.73i)19-s + (−0.707 − 1.22i)20-s + (−1.36 − 0.366i)22-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.803 + 0.595i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ -0.803 + 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4450940525\)
\(L(\frac12)\) \(\approx\) \(0.4450940525\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
37 \( 1 - iT \)
good5 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + iT - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.227103412812027117226131257312, −7.898558966780894714659682041736, −7.31967269346247233106378999626, −6.35673105191577990803343878014, −5.51875786878346871676934854478, −4.61586783497357375636347433454, −4.15118009613352365066664194427, −3.17844772041824238072535592110, −2.34106457220617745367415744036, −0.18336946451761635407621336109, 2.28770609842018944973323650425, 2.84293886470824197643113943516, 3.69651225982350382920569869941, 4.56507130970369862011171768971, 5.27006429574933428098188512629, 6.40401449578399489786428654172, 6.90513711617597845566858625886, 7.53299409268630502713080331222, 8.509992538869177739434304499983, 9.382091246490914476550493193199

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