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Properties

Label 2-2646-9.4-c1-0-28
Degree $2$
Conductor $2646$
Sign $-0.880 + 0.474i$
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 + (−0.794 + 1.37i)5-s + 0.999·8-s + 1.58·10-s + (−0.794 − 1.37i)11-s + (−2.40 + 4.16i)13-s + (−0.5 − 0.866i)16-s + 5.39·17-s − 7.09·19-s + (−0.794 − 1.37i)20-s + (−0.794 + 1.37i)22-s + (0.150 − 0.260i)23-s + (1.23 + 2.14i)25-s + 4.81·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.355 + 0.615i)5-s + 0.353·8-s + 0.502·10-s + (−0.239 − 0.414i)11-s + (−0.667 + 1.15i)13-s + (−0.125 − 0.216i)16-s + 1.30·17-s − 1.62·19-s + (−0.177 − 0.307i)20-s + (−0.169 + 0.293i)22-s + (0.0313 − 0.0542i)23-s + (0.247 + 0.429i)25-s + 0.943·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.880 + 0.474i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.880 + 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3458753937\)
\(L(\frac12)\) \(\approx\) \(0.3458753937\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.794 - 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.794 + 1.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.40 - 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.39T + 17T^{2} \)
19 \( 1 + 7.09T + 19T^{2} \)
23 \( 1 + (-0.150 + 0.260i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.35 - 2.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.88T + 53T^{2} \)
59 \( 1 + (3.23 - 5.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 + 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + (4.19 + 7.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.21T + 89T^{2} \)
97 \( 1 + (0.712 + 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.630360720375438286009716650228, −7.80555717505412601361290629033, −7.18260315610628360764938586663, −6.36076487304893050665711950145, −5.37274774751041833411977561186, −4.29835577989414135198562830530, −3.62101782985632441679691242859, −2.64093695393685550899483372868, −1.76064568658735599323660595464, −0.13990867595718656587888165101, 1.12683147405617377670986172584, 2.49167177381983379697760297021, 3.68633915291335199972451912057, 4.69634672519594938009320101066, 5.30174121882214731047740922744, 6.08945003100878070578586926920, 7.07479335340787160762335225050, 7.77954550245332626408032237230, 8.273616079880888407240544877063, 9.014955758716617077400951944854

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