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Properties

Label 2-2646-9.7-c1-0-0
Degree $2$
Conductor $2646$
Sign $-0.906 - 0.422i$
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 + (1.93 + 3.34i)5-s − 0.999·8-s + 3.86·10-s + (−1.86 + 3.23i)11-s + (−3.34 − 5.79i)13-s + (−0.5 + 0.866i)16-s − 5.41·17-s − 2.96·19-s + (1.93 − 3.34i)20-s + (1.86 + 3.23i)22-s + (−0.732 − 1.26i)23-s + (−4.96 + 8.59i)25-s − 6.69·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.863 + 1.49i)5-s − 0.353·8-s + 1.22·10-s + (−0.562 + 0.974i)11-s + (−0.928 − 1.60i)13-s + (−0.125 + 0.216i)16-s − 1.31·17-s − 0.680·19-s + (0.431 − 0.748i)20-s + (0.397 + 0.689i)22-s + (−0.152 − 0.264i)23-s + (−0.992 + 1.71i)25-s − 1.31·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.906 - 0.422i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2824610033\)
\(L(\frac12)\) \(\approx\) \(0.2824610033\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 3.34i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.86 - 3.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.34 + 5.79i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 + 2.96T + 19T^{2} \)
23 \( 1 + (0.732 + 1.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.896 + 1.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.535T + 37T^{2} \)
41 \( 1 + (0.637 + 1.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.86 - 3.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.27 + 9.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.92T + 53T^{2} \)
59 \( 1 + (4.31 + 7.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.48 - 6.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.76 + 4.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + 6.83T + 73T^{2} \)
79 \( 1 + (-2.46 + 4.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.95 - 15.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + (-2.94 + 5.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.574561371991877491296204832487, −8.496720219760658030331470105179, −7.41880970457059544708140571057, −6.91806850617042504344356826466, −6.02284862111188440633053835623, −5.30661259627762050408821434425, −4.43455129530864611982071996938, −3.25172321820602094673481521430, −2.50328272933896766731377753227, −2.01315004944959364625017501023, 0.07095824895457260543716385948, 1.68586049708528770384364895035, 2.59186221571628398739512422838, 4.22872081607499093208044891056, 4.55927519368810514731502984619, 5.46752460545639044315292510904, 6.08275805207656798495479558244, 6.85833473542704846314411127727, 7.82859547118256878659844455259, 8.724296826846589891789188042102

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