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Properties

Label 2-2646-63.38-c1-0-6
Degree $2$
Conductor $2646$
Sign $-0.881 + 0.472i$
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  + i·2-s − 4-s + (−0.895 + 1.55i)5-s − i·8-s + (−1.55 − 0.895i)10-s + (−2.07 + 1.20i)11-s + (4.23 − 2.44i)13-s + 16-s + (−1.83 + 3.17i)17-s + (−2.61 + 1.50i)19-s + (0.895 − 1.55i)20-s + (−1.20 − 2.07i)22-s + (3.26 + 1.88i)23-s + (0.897 + 1.55i)25-s + (2.44 + 4.23i)26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.400 + 0.693i)5-s − 0.353i·8-s + (−0.490 − 0.283i)10-s + (−0.627 + 0.362i)11-s + (1.17 − 0.678i)13-s + 0.250·16-s + (−0.444 + 0.769i)17-s + (−0.599 + 0.346i)19-s + (0.200 − 0.346i)20-s + (−0.256 − 0.443i)22-s + (0.680 + 0.392i)23-s + (0.179 + 0.310i)25-s + (0.479 + 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.881 + 0.472i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.881 + 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7013128904\)
\(L(\frac12)\) \(\approx\) \(0.7013128904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.895 - 1.55i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.07 - 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.23 + 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.83 - 3.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.61 - 1.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.26 - 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.68 - 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.64iT - 31T^{2} \)
37 \( 1 + (4.68 + 8.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.04 + 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.48 - 6.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.13T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 14.5T + 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 - 0.570T + 67T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 + (10.7 + 6.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 + (7.00 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.87 - 3.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.77 + 2.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.931308134250936567196141661408, −8.514321896230557663818010629363, −7.69944749153965631944106050211, −7.01894431330821819239454918923, −6.33847436845701141831350040702, −5.55083040721690255292912719753, −4.68989378534869854414342656314, −3.66813022533517392133791399309, −3.00856960815784901994668015615, −1.51284847288550778924306920809, 0.24192702416259184871376479190, 1.38422791547186647170477928762, 2.60060551326828825645289973306, 3.47250288887474356758168533060, 4.59041664972931216982129648803, 4.83879464405825966583204338525, 6.12639281404645085686681234358, 6.80139860635577435238745865536, 8.035670761498380395199937937959, 8.534006754703433558211871465212

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