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Properties

Label 2-2646-63.16-c1-0-9
Degree $2$
Conductor $2646$
Sign $0.823 - 0.566i$
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 − 3·5-s + 0.999·8-s + (1.5 + 2.59i)10-s + 6·11-s + (1 + 1.73i)13-s + (−0.5 − 0.866i)16-s + (−3 − 5.19i)17-s + (−3.5 + 6.06i)19-s + (1.49 − 2.59i)20-s + (−3 − 5.19i)22-s − 3·23-s + 4·25-s + (0.999 − 1.73i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.34·5-s + 0.353·8-s + (0.474 + 0.821i)10-s + 1.80·11-s + (0.277 + 0.480i)13-s + (−0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + (−0.802 + 1.39i)19-s + (0.335 − 0.580i)20-s + (−0.639 − 1.10i)22-s − 0.625·23-s + 0.800·25-s + (0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.823 - 0.566i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.823 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8510696998\)
\(L(\frac12)\) \(\approx\) \(0.8510696998\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.875789891962849293403792378435, −8.343897555668474574063861872666, −7.54371351064716783115683329774, −6.77739746049386281226645157403, −6.02889703033778877596416487259, −4.41876550517064804341344394664, −4.17149915107416829578982286192, −3.38820515104997372048400013766, −2.12761041558773500646144942325, −0.941059663090886205703040339871, 0.41718385293293338696864043095, 1.72859897867610761726232932864, 3.35198720576163006533059845225, 4.12028699536755786353381521402, 4.66501747363712042326041385704, 5.97183194512435554111946878815, 6.69464132667880455127242339186, 7.14040562688200470193188436231, 8.214070908011174031726282009713, 8.622826064219430318628867187726

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