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Properties

Label 2-820-41.23-c1-0-2
Degree $2$
Conductor $820$
Sign $-0.830 - 0.557i$
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  + 1.12i·3-s + (0.809 − 0.587i)5-s + (−3.16 − 1.02i)7-s + 1.74·9-s + (−2.39 + 3.30i)11-s + (−2.71 + 0.882i)13-s + (0.659 + 0.907i)15-s + (−1.96 + 2.70i)17-s + (−5.33 − 1.73i)19-s + (1.15 − 3.54i)21-s + (2.70 + 8.32i)23-s + (0.309 − 0.951i)25-s + 5.31i·27-s + (−1.20 − 1.65i)29-s + (2.94 + 2.13i)31-s + ⋯
L(s)  = 1  + 0.647i·3-s + (0.361 − 0.262i)5-s + (−1.19 − 0.388i)7-s + 0.580·9-s + (−0.723 + 0.995i)11-s + (−0.753 + 0.244i)13-s + (0.170 + 0.234i)15-s + (−0.476 + 0.655i)17-s + (−1.22 − 0.397i)19-s + (0.251 − 0.773i)21-s + (0.564 + 1.73i)23-s + (0.0618 − 0.190i)25-s + 1.02i·27-s + (−0.223 − 0.307i)29-s + (0.528 + 0.384i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.830 - 0.557i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.830 - 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216239 + 0.709835i\)
\(L(\frac12)\) \(\approx\) \(0.216239 + 0.709835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (4.48 - 4.57i)T \)
good3 \( 1 - 1.12iT - 3T^{2} \)
7 \( 1 + (3.16 + 1.02i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (2.39 - 3.30i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.71 - 0.882i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.96 - 2.70i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.33 + 1.73i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.70 - 8.32i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.20 + 1.65i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.94 - 2.13i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.36 + 2.44i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-1.21 - 3.74i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (11.6 - 3.79i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.30 - 5.92i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.53 + 13.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.60 + 4.92i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.33 + 3.21i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-9.50 + 13.0i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 14.5iT - 79T^{2} \)
83 \( 1 + 6.08T + 83T^{2} \)
89 \( 1 + (-12.6 - 4.12i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (9.30 + 12.8i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.34340033834664357015138268364, −9.620428640977642850324932900724, −9.400849167153307176345463666518, −7.992920689023523830621874047409, −7.03280952958093838000725004884, −6.33636097293277058390406961351, −5.01209438860455558367166906167, −4.37414870047137612014149929559, −3.25013468276420068879889103129, −1.90072378746532522506870181724, 0.33903368494838817264833332357, 2.28901531489021517197761542279, 3.01899488901494349996082909167, 4.47925012403123553301528903371, 5.70367301590361767092728749383, 6.53708945768375235113784077831, 7.05598791260307228197569183056, 8.234153488392187215559621645691, 8.974669126335825747993434497758, 10.11790560550452936439845243545

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