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Properties

Label 2-820-164.11-c1-0-28
Degree $2$
Conductor $820$
Sign $0.242 - 0.970i$
Analytic cond. $6.54773$
Root an. cond. $2.55885$
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  + (−0.00265 + 1.41i)2-s + (0.239 − 0.578i)3-s + (−1.99 − 0.00749i)4-s + (−0.453 + 0.891i)5-s + (0.817 + 0.340i)6-s + (−0.481 − 2.00i)7-s + (0.0159 − 2.82i)8-s + (1.84 + 1.84i)9-s + (−1.25 − 0.644i)10-s + (0.379 − 0.444i)11-s + (−0.483 + 1.15i)12-s + (−2.37 − 3.88i)13-s + (2.83 − 0.675i)14-s + (0.406 + 0.476i)15-s + (3.99 + 0.0299i)16-s + (0.493 + 6.26i)17-s + ⋯
L(s)  = 1  + (−0.00187 + 0.999i)2-s + (0.138 − 0.334i)3-s + (−0.999 − 0.00374i)4-s + (−0.203 + 0.398i)5-s + (0.333 + 0.138i)6-s + (−0.181 − 0.758i)7-s + (0.00562 − 0.999i)8-s + (0.614 + 0.614i)9-s + (−0.398 − 0.203i)10-s + (0.114 − 0.133i)11-s + (−0.139 + 0.333i)12-s + (−0.659 − 1.07i)13-s + (0.758 − 0.180i)14-s + (0.105 + 0.122i)15-s + (0.999 + 0.00749i)16-s + (0.119 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13255 + 0.884045i\)
\(L(\frac12)\) \(\approx\) \(1.13255 + 0.884045i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.00265 - 1.41i)T \)
5 \( 1 + (0.453 - 0.891i)T \)
41 \( 1 + (-6.28 + 1.20i)T \)
good3 \( 1 + (-0.239 + 0.578i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.481 + 2.00i)T + (-6.23 + 3.17i)T^{2} \)
11 \( 1 + (-0.379 + 0.444i)T + (-1.72 - 10.8i)T^{2} \)
13 \( 1 + (2.37 + 3.88i)T + (-5.90 + 11.5i)T^{2} \)
17 \( 1 + (-0.493 - 6.26i)T + (-16.7 + 2.65i)T^{2} \)
19 \( 1 + (-7.07 - 4.33i)T + (8.62 + 16.9i)T^{2} \)
23 \( 1 + (-5.02 - 3.64i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.163 - 2.07i)T + (-28.6 - 4.53i)T^{2} \)
31 \( 1 + (-2.10 + 6.46i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.846 - 2.60i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (0.399 + 2.52i)T + (-40.8 + 13.2i)T^{2} \)
47 \( 1 + (0.607 - 2.52i)T + (-41.8 - 21.3i)T^{2} \)
53 \( 1 + (11.5 + 0.907i)T + (52.3 + 8.29i)T^{2} \)
59 \( 1 + (4.45 - 6.13i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.25 - 14.2i)T + (-58.0 - 18.8i)T^{2} \)
67 \( 1 + (-11.9 + 10.2i)T + (10.4 - 66.1i)T^{2} \)
71 \( 1 + (-7.32 - 6.25i)T + (11.1 + 70.1i)T^{2} \)
73 \( 1 + (1.12 - 1.12i)T - 73iT^{2} \)
79 \( 1 + (0.907 + 0.376i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 - 3.08iT - 83T^{2} \)
89 \( 1 + (1.34 - 0.321i)T + (79.2 - 40.4i)T^{2} \)
97 \( 1 + (1.23 - 1.05i)T + (15.1 - 95.8i)T^{2} \)
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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

−10.16306447512493157035513421924, −9.665563069124912069478486330422, −8.304900330855338198019364360524, −7.57315181157992028518770020516, −7.31344234521454962769715913434, −6.12752995490637771055952040025, −5.28147882024887806666305691898, −4.13120115954836963480670030795, −3.19936267861023929652136661460, −1.17477301305535294078965374596, 0.918358842412623481442098443766, 2.51190186705407586843544641950, 3.38761760831782487986865469069, 4.76145414519322633995601563900, 4.98474251759348712599341686680, 6.63617500731900592496566478842, 7.58721749330775357152046578256, 8.945416718722286569528095413518, 9.365773974695339541092517687723, 9.726582549434830323787610158696

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