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Properties

Label 2-2646-63.5-c1-0-27
Degree $2$
Conductor $2646$
Sign $-0.334 + 0.942i$
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 + (0.183 + 0.317i)5-s + i·8-s + (0.317 − 0.183i)10-s + (0.579 + 0.334i)11-s + (−0.867 − 0.500i)13-s + 16-s + (−2.49 − 4.32i)17-s + (5.50 + 3.17i)19-s + (−0.183 − 0.317i)20-s + (0.334 − 0.579i)22-s + (−6.66 + 3.84i)23-s + (2.43 − 4.21i)25-s + (−0.500 + 0.867i)26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.0819 + 0.141i)5-s + 0.353i·8-s + (0.100 − 0.0579i)10-s + (0.174 + 0.100i)11-s + (−0.240 − 0.138i)13-s + 0.250·16-s + (−0.605 − 1.04i)17-s + (1.26 + 0.729i)19-s + (−0.0409 − 0.0709i)20-s + (0.0713 − 0.123i)22-s + (−1.38 + 0.802i)23-s + (0.486 − 0.842i)25-s + (−0.0982 + 0.170i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.334 + 0.942i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.334 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.432425237\)
\(L(\frac12)\) \(\approx\) \(1.432425237\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.183 - 0.317i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.579 - 0.334i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.867 + 0.500i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.49 + 4.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.50 - 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.66 - 3.84i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.58 - 0.914i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.15 + 3.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.32T + 47T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.72T + 59T^{2} \)
61 \( 1 + 4.95iT - 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 5.49iT - 71T^{2} \)
73 \( 1 + (-3.52 + 2.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.35T + 79T^{2} \)
83 \( 1 + (8.50 + 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.35 + 9.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.9 - 8.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.862310180727632071047345242023, −7.76539417806238993814220071074, −7.37053607173676608831084163002, −6.14856298806015361886659864488, −5.49144330506181782607866261323, −4.50611505533792737979852220149, −3.74268119512084984884762096058, −2.75389886850296791071322643255, −1.90183364567740351057646897126, −0.53793093468136486285551161536, 1.10611210317967568269534558176, 2.45592012773132208879602867103, 3.64669981285146194627548643492, 4.46212533814801181111365093473, 5.30031519887754388173588981313, 6.05653711315156068165462586512, 6.82695511107916981030595446036, 7.49243112963623035008001672515, 8.367514451019911878753498121111, 8.917397834696519762755214984857

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