Properties

Label 2-2646-9.4-c1-0-27
Degree $2$
Conductor $2646$
Sign $0.851 + 0.524i$
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 + (−1.84 + 3.20i)5-s − 0.999·8-s − 3.69·10-s + (−0.738 − 1.27i)11-s + (1.34 − 2.33i)13-s + (−0.5 − 0.866i)16-s − 6.57·17-s − 0.888·19-s + (−1.84 − 3.20i)20-s + (0.738 − 1.27i)22-s + (3.14 − 5.44i)23-s + (−4.34 − 7.52i)25-s + 2.69·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.827 + 1.43i)5-s − 0.353·8-s − 1.16·10-s + (−0.222 − 0.385i)11-s + (0.374 − 0.648i)13-s + (−0.125 − 0.216i)16-s − 1.59·17-s − 0.203·19-s + (−0.413 − 0.716i)20-s + (0.157 − 0.272i)22-s + (0.655 − 1.13i)23-s + (−0.868 − 1.50i)25-s + 0.529·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.851 + 0.524i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7670266553\)
\(L(\frac12)\) \(\approx\) \(0.7670266553\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 - 3.20i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.738 + 1.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.34 + 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.57T + 17T^{2} \)
19 \( 1 + 0.888T + 19T^{2} \)
23 \( 1 + (-3.14 + 5.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.40 + 5.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.77T + 37T^{2} \)
41 \( 1 + (2.05 - 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.00618 - 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.49 - 6.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 + (3.45 - 5.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.86 + 4.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.73 + 8.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 + (5.72 + 9.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.23 - 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.87T + 89T^{2} \)
97 \( 1 + (-6.58 - 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.488731853402373594114523073725, −7.913266596411233612462865585835, −7.21848790791921850721939890938, −6.41625589047958631050240470481, −6.07850103107823092022332007457, −4.72317835640381437460982402249, −4.07292421425174863511316567206, −3.10053816921755083762290948468, −2.48165559583459091655165003788, −0.24522628577142820703704552025, 1.11866734997559961697648318597, 2.08500063811174996808168102549, 3.42336300703873058217754732105, 4.27275974145317173924681500166, 4.76003152706565313714558018273, 5.49960295180897441354906766931, 6.65127783229098613886299621350, 7.43580897166223936412347302030, 8.511207432432065593885997976773, 8.860383961155017218395673946980

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