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Properties

Label 2-2646-63.4-c1-0-1
Degree $2$
Conductor $2646$
Sign $0.130 - 0.991i$
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 − 3.44·5-s − 0.999·8-s + (−1.72 + 2.98i)10-s − 2·11-s + (2.44 − 4.24i)13-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−3.72 − 6.45i)19-s + (1.72 + 2.98i)20-s + (−1 + 1.73i)22-s + 23-s + 6.89·25-s + (−2.44 − 4.24i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.54·5-s − 0.353·8-s + (−0.545 + 0.944i)10-s − 0.603·11-s + (0.679 − 1.17i)13-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.854 − 1.48i)19-s + (0.385 + 0.667i)20-s + (−0.213 + 0.369i)22-s + 0.208·23-s + 1.37·25-s + (−0.480 − 0.832i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.130 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1566360708\)
\(L(\frac12)\) \(\approx\) \(0.1566360708\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.44T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.72 + 6.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + (-1.44 - 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 - 2.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.89 - 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.550 - 0.953i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.72 - 9.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.55 - 2.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 + (1.44 - 2.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.94 - 6.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.55 + 6.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.44 - 5.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.940033723339958849083999901796, −8.215134200942930806696543644984, −7.69722224195739754434420481050, −6.79122354035307239535615442110, −5.81545162879195288271768667267, −4.78718198808210248617648310748, −4.32433601038160921969752134919, −3.18743402880307755030931399747, −2.80526343340802487095454229029, −1.03850406949636762748391715834, 0.05383231886163351659597374867, 1.87639930386285147495056142462, 3.42811137801925666191954089918, 3.89472952762302052476656000326, 4.60739259439891195816549691231, 5.62019106833433941663462205394, 6.46741383092908585713269574346, 7.22104762358732663527054568475, 7.898908002510045402934185142226, 8.463106610629964122942584279194

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