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Properties

Label 2-2646-63.38-c1-0-39
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

\(L(1)\) \(\approx\) \(1.541585999\)
\(L(\frac12)\) \(\approx\) \(1.541585999\)
\(L(\frac{3}{2})\) 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

−8.630656276581206084539722135848, −8.154926380166336699508851223591, −6.79642659176732772806451242237, −5.96143043778500804604611267520, −5.25291534480596877724170817533, −4.41363639529242710054073558873, −3.77535304454630415563929694037, −2.29695228638297342700187821481, −1.60929330395034049353390662070, −0.48590678720791506874408313476, 1.67040151558491165231775582482, 2.69364520404514024022453452297, 3.61673040299888404919519842368, 4.65481391460417952996846383031, 5.60359249130197295427424748787, 6.43587160980736389385132954106, 6.77672780691265227281051433650, 7.51019175927600822241552108270, 8.357317095177749219058530210383, 9.426460370314220905050086333877

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