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Properties

Label 2-2646-63.38-c1-0-19
Degree $2$
Conductor $2646$
Sign $0.997 + 0.0729i$
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 + (−1.94 + 3.36i)5-s + i·8-s + (3.36 + 1.94i)10-s + (3.41 − 1.97i)11-s + (2.46 − 1.42i)13-s + 16-s + (0.371 − 0.642i)17-s + (1.54 − 0.892i)19-s + (1.94 − 3.36i)20-s + (−1.97 − 3.41i)22-s + (−5.41 − 3.12i)23-s + (−5.07 − 8.78i)25-s + (−1.42 − 2.46i)26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.870 + 1.50i)5-s + 0.353i·8-s + (1.06 + 0.615i)10-s + (1.03 − 0.594i)11-s + (0.684 − 0.395i)13-s + 0.250·16-s + (0.0899 − 0.155i)17-s + (0.354 − 0.204i)19-s + (0.435 − 0.753i)20-s + (−0.420 − 0.728i)22-s + (−1.12 − 0.651i)23-s + (−1.01 − 1.75i)25-s + (−0.279 − 0.483i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.997 + 0.0729i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.997 + 0.0729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.475414355\)
\(L(\frac12)\) \(\approx\) \(1.475414355\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.94 - 3.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.41 + 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.46 + 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.371 + 0.642i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.54 + 0.892i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.41 + 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.50 - 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.51iT - 31T^{2} \)
37 \( 1 + (1.50 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.24 - 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 + 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.18T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.0211T + 59T^{2} \)
61 \( 1 - 2.46iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + (-4.20 - 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.63T + 79T^{2} \)
83 \( 1 + (-4.02 + 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.63 + 8.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.2 - 9.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.820301502945655151366755654668, −8.153029983458322394046786858122, −7.37986758876034373794224800454, −6.51024810815999420037890570410, −5.96907391170473857489915096521, −4.60882643260953573958583475187, −3.67493883919064882058288843610, −3.30201880545546331091232867552, −2.31752821302550113300675178437, −0.840777804779361973876499413009, 0.73366551233809028703707592696, 1.77563351607136768769200904371, 3.80442852635439281905706896223, 4.04581927887046061795359599239, 4.94928615149995755671562668980, 5.75459742681225378449067374733, 6.57820513340057552892480686719, 7.53510373503724602472158902769, 8.072906726668832311335451150445, 8.796079584518851068254461753817

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