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Properties

Label 2-2646-63.4-c1-0-7
Degree $2$
Conductor $2646$
Sign $0.641 - 0.766i$
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 − 2.03·5-s − 0.999·8-s + (−1.01 + 1.75i)10-s − 4·11-s + (2.12 − 3.67i)13-s + (−0.5 + 0.866i)16-s + (0.707 − 1.22i)17-s + (0.398 + 0.690i)19-s + (1.01 + 1.75i)20-s + (−2 + 3.46i)22-s − 6.74·23-s − 0.872·25-s + (−2.12 − 3.67i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.908·5-s − 0.353·8-s + (−0.321 + 0.556i)10-s − 1.20·11-s + (0.588 − 1.01i)13-s + (−0.125 + 0.216i)16-s + (0.171 − 0.297i)17-s + (0.0914 + 0.158i)19-s + (0.227 + 0.393i)20-s + (−0.426 + 0.738i)22-s − 1.40·23-s − 0.174·25-s + (−0.416 − 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.641 - 0.766i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.641 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7638947640\)
\(L(\frac12)\) \(\approx\) \(0.7638947640\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.03T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-2.12 + 3.67i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.398 - 0.690i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + (-4.43 - 7.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.73 - 4.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.87 - 8.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.82 + 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.43 + 7.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.44 + 5.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.30 - 9.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.32 - 2.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.398 - 0.690i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.436 + 0.756i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 + (7.68 - 13.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.93 - 5.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.15 - 7.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.92 - 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.848187222799957900829272892723, −8.154855922185027637465379386204, −7.68062430234393041462705595310, −6.63289207179348900556388824262, −5.60922562217708273001988168476, −5.03225963146508036724302054903, −4.03574862308604400112838344251, −3.28718134838937234330235257830, −2.51015281119501595963112494287, −1.04934126525059008811751989069, 0.25444873805215802247227216601, 2.11821391935718285606820694945, 3.22605811437590871648727462211, 4.25569810971528458662480375049, 4.55996754335371110265987541313, 5.92222134978643757851395621790, 6.23695607490035819511232414935, 7.43693441589443550778057089471, 7.935550626701999116810958780453, 8.353880345909144581190755271908

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