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Properties

Label 2-2646-63.16-c1-0-30
Degree $2$
Conductor $2646$
Sign $0.940 + 0.339i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.460·5-s − 0.999·8-s + (−0.230 − 0.398i)10-s + 3.64·11-s + (−0.730 − 1.26i)13-s + (−0.5 − 0.866i)16-s + (−1.86 − 3.23i)17-s + (2.02 − 3.51i)19-s + (0.230 − 0.398i)20-s + (1.82 + 3.15i)22-s − 1.13·23-s − 4.78·25-s + (0.730 − 1.26i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.205·5-s − 0.353·8-s + (−0.0728 − 0.126i)10-s + 1.09·11-s + (−0.202 − 0.350i)13-s + (−0.125 − 0.216i)16-s + (−0.452 − 0.784i)17-s + (0.465 − 0.805i)19-s + (0.0514 − 0.0891i)20-s + (0.388 + 0.673i)22-s − 0.236·23-s − 0.957·25-s + (0.143 − 0.248i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.774754919\)
\(L(\frac12)\) \(\approx\) \(1.774754919\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.460T + 5T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + (0.730 + 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.13T + 23T^{2} \)
29 \( 1 + (-4.48 + 7.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.257 - 0.445i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.472 + 0.819i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.66 + 8.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.16 + 2.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.21 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.04 - 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 + (-6.62 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.32 + 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.36 - 2.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.59 + 9.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.625242499717903570976606634372, −8.066845881547693980395178482426, −7.06364352105125311437252327187, −6.68628341230708617116766900141, −5.73292664491964960525078531736, −4.91263456825375459694645984792, −4.14079198770771820368563678717, −3.30698184618755927877814865014, −2.18807736357311127855259370649, −0.56034531404794795838742398385, 1.21988274899989367409107153294, 2.11067133706490015377796716405, 3.38307198855084322250640505783, 3.98938763013626370649639004378, 4.77284085007911669632767663643, 5.83191361388010338443333195134, 6.44333423787531425871371372199, 7.36310463533160068096847013908, 8.244459611426932032905987255384, 9.096295080884274008725176497046

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