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Properties

Label 2-2646-63.25-c1-0-32
Degree $2$
Conductor $2646$
Sign $0.732 + 0.681i$
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 + (1.62 − 2.82i)5-s + 8-s + (1.62 − 2.82i)10-s + (2.81 + 4.87i)11-s + (−0.613 − 1.06i)13-s + 16-s + (2.95 − 5.11i)17-s + (1.32 + 2.29i)19-s + (1.62 − 2.82i)20-s + (2.81 + 4.87i)22-s + (3.31 − 5.73i)23-s + (−2.81 − 4.87i)25-s + (−0.613 − 1.06i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.728 − 1.26i)5-s + 0.353·8-s + (0.515 − 0.892i)10-s + (0.847 + 1.46i)11-s + (−0.170 − 0.294i)13-s + 0.250·16-s + (0.716 − 1.24i)17-s + (0.303 + 0.525i)19-s + (0.364 − 0.631i)20-s + (0.599 + 1.03i)22-s + (0.690 − 1.19i)23-s + (−0.562 − 0.974i)25-s + (−0.120 − 0.208i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.732 + 0.681i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.732 + 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.625947697\)
\(L(\frac12)\) \(\approx\) \(3.625947697\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.82i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.81 - 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.613 + 1.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 5.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.31 + 5.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.95 - 5.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.81 - 6.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 - 4.48T + 61T^{2} \)
67 \( 1 + 5.62T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + (-5.59 + 9.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 + (-3.87 + 6.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.48 + 7.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.53 + 2.65i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.067869273739476848468308520667, −7.85319395801063759823983562324, −7.19081795294882950606759461579, −6.33723146519222588597726963598, −5.42685567451482062262330871203, −4.86086387196301912240819588220, −4.28359178822620502633177873419, −3.05355521421765639707123096477, −1.94685211656325979299094268365, −1.06490544794375374870305826125, 1.35392953372212519608363938723, 2.47041748646888862525483567180, 3.41685446740096527117448335716, 3.82930412875354708168237370426, 5.32180260333931076534690654882, 5.86912151090727035642045090224, 6.52657114075377170646936399149, 7.13777689960191540851341876946, 8.068830073007689064274084508147, 9.046825890280262004077036214469

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