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Properties

Label 2-2646-63.25-c1-0-3
Degree $2$
Conductor $2646$
Sign $-0.841 - 0.540i$
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  − 2-s + 4-s + (−0.5 + 0.866i)5-s − 8-s + (0.5 − 0.866i)10-s + (−1 − 1.73i)11-s + (1 + 1.73i)13-s + 16-s + (−3.5 − 6.06i)19-s + (−0.5 + 0.866i)20-s + (1 + 1.73i)22-s + (1.5 − 2.59i)23-s + (2 + 3.46i)25-s + (−1 − 1.73i)26-s + (−4 + 6.92i)29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.223 + 0.387i)5-s − 0.353·8-s + (0.158 − 0.273i)10-s + (−0.301 − 0.522i)11-s + (0.277 + 0.480i)13-s + 0.250·16-s + (−0.802 − 1.39i)19-s + (−0.111 + 0.193i)20-s + (0.213 + 0.369i)22-s + (0.312 − 0.541i)23-s + (0.400 + 0.692i)25-s + (−0.196 − 0.339i)26-s + (−0.742 + 1.28i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3835573829\)
\(L(\frac12)\) \(\approx\) \(0.3835573829\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.958936850180151200047464674130, −8.639501344140736992176898928331, −7.63782663814157132700011162854, −6.96053193628730835477316669173, −6.37867537589474887618273869676, −5.36653214335777075098319266570, −4.41874275029269925725293924620, −3.29446974590909965284528332835, −2.54336129721404643774640162448, −1.27891257906430064975896778948, 0.16518598319650322148658524311, 1.53731571314976704990743780844, 2.51204704825734246898830500316, 3.71912180387548680944158924635, 4.50225298417750613081317046707, 5.68957821172177873171104466655, 6.16224845467673619849764551629, 7.40551320536375770118039094967, 7.76133163993439283225952703900, 8.553103599334551824871511728504

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