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Properties

Label 2-2646-9.4-c1-0-3
Degree $2$
Conductor $2646$
Sign $-0.254 + 0.967i$
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 + (−1.72 + 2.98i)5-s − 0.999·8-s − 3.44·10-s + (1 + 1.73i)11-s + (−2.44 + 4.24i)13-s + (−0.5 − 0.866i)16-s + 2·17-s − 7.44·19-s + (−1.72 − 2.98i)20-s + (−0.999 + 1.73i)22-s + (−0.5 + 0.866i)23-s + (−3.44 − 5.97i)25-s − 4.89·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.771 + 1.33i)5-s − 0.353·8-s − 1.09·10-s + (0.301 + 0.522i)11-s + (−0.679 + 1.17i)13-s + (−0.125 − 0.216i)16-s + 0.485·17-s − 1.70·19-s + (−0.385 − 0.667i)20-s + (−0.213 + 0.369i)22-s + (−0.104 + 0.180i)23-s + (−0.689 − 1.19i)25-s − 0.960·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.254 + 0.967i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.254 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6526353762\)
\(L(\frac12)\) \(\approx\) \(0.6526353762\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
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 \( 1 + 7.79T + 37T^{2} \)
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 \( 1 - 1.10T + 53T^{2} \)
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 \( 1 + 2.89T + 73T^{2} \)
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 \( 1 + 7.10T + 89T^{2} \)
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

−9.263198160531840258785004434166, −8.496402641911257528549720691618, −7.55386890413707584400677132827, −7.08666808350214342163489339144, −6.54796287543131617900196837289, −5.72138455920183828080322760480, −4.36401956807987790191240908321, −4.12679973532163535094640997622, −2.98329565496159189623379596565, −2.07596865432345633931439009166, 0.20333762234735414125880753994, 1.15904113096762294727840466839, 2.49001447018966281721465186996, 3.55848355867994167579369832134, 4.30678077884985918323121468825, 5.02210347798528160777369122748, 5.69997338213311447469964492662, 6.71939565173105495520712960639, 7.85225694538208931205805378367, 8.456422693191094016505632035184

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