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Properties

Label 2-2664-2664.787-c0-0-0
Degree $2$
Conductor $2664$
Sign $0.200 - 0.979i$
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  + 2-s − 3-s + 4-s + 2i·5-s − 6-s + (0.866 + 0.5i)7-s + 8-s + 9-s + 2i·10-s + (0.5 + 0.866i)11-s − 12-s + (0.866 + 0.5i)14-s − 2i·15-s + 16-s + (0.5 − 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 2i·5-s − 6-s + (0.866 + 0.5i)7-s + 8-s + 9-s + 2i·10-s + (0.5 + 0.866i)11-s − 12-s + (0.866 + 0.5i)14-s − 2i·15-s + 16-s + (0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.200 - 0.979i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ 0.200 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.971597268\)
\(L(\frac12)\) \(\approx\) \(1.971597268\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.576651865487779115605867005167, −7.984838143090963376315065702421, −7.27004219840947574697427896619, −6.74080170838988257635323036004, −6.19943840523013521737338107274, −5.34574614761706370754907764372, −4.53901994322779646355280884674, −3.73203585354155122177891479131, −2.57006299835687468878428135285, −1.92236458969823212868425457315, 1.22097630734855354938861621211, 1.64559591116499920025554879715, 3.81585275138086549106517932102, 4.21994243043075635112543811657, 4.96506311189518886704310911351, 5.75880407151156732546944751865, 5.99145571230371055501214239233, 7.31278641324131355938514636552, 8.063048165179789545291240643804, 8.647561612074441355826861530655

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