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Properties

Label 2-2646-63.16-c1-0-13
Degree $2$
Conductor $2646$
Sign $0.873 + 0.487i$
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 − 1.93·5-s + 0.999·8-s + (0.965 + 1.67i)10-s − 5.46·11-s + (1.22 + 2.12i)13-s + (−0.5 − 0.866i)16-s + (−3.15 − 5.46i)17-s + (−0.965 + 1.67i)19-s + (0.965 − 1.67i)20-s + (2.73 + 4.73i)22-s + 5.92·23-s − 1.26·25-s + (1.22 − 2.12i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.863·5-s + 0.353·8-s + (0.305 + 0.529i)10-s − 1.64·11-s + (0.339 + 0.588i)13-s + (−0.125 − 0.216i)16-s + (−0.765 − 1.32i)17-s + (−0.221 + 0.383i)19-s + (0.215 − 0.374i)20-s + (0.582 + 1.00i)22-s + 1.23·23-s − 0.253·25-s + (0.240 − 0.416i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.873 + 0.487i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.873 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7739272629\)
\(L(\frac12)\) \(\approx\) \(0.7739272629\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93T + 5T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.15 + 5.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.965 - 1.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.92T + 23T^{2} \)
29 \( 1 + (0.366 - 0.633i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.67 - 6.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.82 + 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.901 + 1.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.76 - 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.63 - 2.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.50 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.48 + 2.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.09 + 12.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + (-4.76 - 8.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.06 - 8.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.94 + 8.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.64 - 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.93 + 3.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.922373147844951496445968720300, −8.039752135510350852089350269451, −7.45803757972636129193632760132, −6.80460337369196922846627258157, −5.45612282288869084523409732175, −4.77378564313424118604515623702, −3.87609267803137715851223536615, −2.96248830534018031595033740979, −2.14064137918624919859707316315, −0.58319920131682102901245747328, 0.53614445913597701884539483424, 2.17205070070983240598044080629, 3.29862487261348060001989788854, 4.28058538657993100866328968187, 5.10430538742795413853852523029, 5.87599010802611889614189673400, 6.74902351058059415696085464015, 7.58600826995688377522626899497, 8.153074432481093104839684461014, 8.558599293221654038197322752145

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