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Properties

Label 2-2646-21.20-c1-0-36
Degree $2$
Conductor $2646$
Sign $0.912 - 0.409i$
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 + 3.57·5-s − i·8-s + 3.57i·10-s − 5.52i·11-s + 6.91i·13-s + 16-s + 2.44·17-s − 8.16i·19-s − 3.57·20-s + 5.52·22-s − 3.08i·23-s + 7.81·25-s − 6.91·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.60·5-s − 0.353i·8-s + 1.13i·10-s − 1.66i·11-s + 1.91i·13-s + 0.250·16-s + 0.594·17-s − 1.87i·19-s − 0.800·20-s + 1.17·22-s − 0.644i·23-s + 1.56·25-s − 1.35·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.912 - 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.423550899\)
\(L(\frac12)\) \(\approx\) \(2.423550899\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.57T + 5T^{2} \)
11 \( 1 + 5.52iT - 11T^{2} \)
13 \( 1 - 6.91iT - 13T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 + 8.16iT - 19T^{2} \)
23 \( 1 + 3.08iT - 23T^{2} \)
29 \( 1 - 5.66iT - 29T^{2} \)
31 \( 1 + 0.400iT - 31T^{2} \)
37 \( 1 - 5.79T + 37T^{2} \)
41 \( 1 - 5.49T + 41T^{2} \)
43 \( 1 + 2.95T + 43T^{2} \)
47 \( 1 - 0.0958T + 47T^{2} \)
53 \( 1 + 5.17iT - 53T^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 + 7.06iT - 61T^{2} \)
67 \( 1 + 1.45T + 67T^{2} \)
71 \( 1 - 8.77iT - 71T^{2} \)
73 \( 1 - 0.679iT - 73T^{2} \)
79 \( 1 + 6.43T + 79T^{2} \)
83 \( 1 - 1.64T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−9.021242090966055531329705370310, −8.354986820275058671731669696053, −7.06523188203429647386448454362, −6.55881920572911580755112733317, −5.94906036674762096686195164987, −5.21053285336431137308549667015, −4.41045164686955933125664994117, −3.14890670093741767359747920966, −2.16761998377377945853474616946, −0.922432925687256359193088247996, 1.18107632009353658200254366004, 2.03456014349851759363518623196, 2.82626858223260429041409066152, 3.89968854425356961028498666363, 4.99650780944504398318539139874, 5.68212296123991361381034462445, 6.16703864652225570670117981953, 7.55151362659175863031077557323, 7.964921935918498134315229056289, 9.153717034665947467403927546238

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