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Properties

Label 2-820-820.219-c1-0-99
Degree $2$
Conductor $820$
Sign $-0.274 + 0.961i$
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 − i)2-s − 2i·4-s + (2.12 − 0.707i)5-s + (−2 − 2i)8-s + (2.12 − 2.12i)9-s + (1.41 − 2.82i)10-s + (−1.29 + 0.535i)13-s − 4·16-s + (1.12 + 0.464i)17-s − 4.24i·18-s + (−1.41 − 4.24i)20-s + (3.99 − 3i)25-s + (−0.757 + 1.82i)26-s + (−0.0502 − 0.121i)29-s + (−4 + 4i)32-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (0.948 − 0.316i)5-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (0.447 − 0.894i)10-s + (−0.358 + 0.148i)13-s − 16-s + (0.271 + 0.112i)17-s − 0.999i·18-s + (−0.316 − 0.948i)20-s + (0.799 − 0.600i)25-s + (−0.148 + 0.358i)26-s + (−0.00933 − 0.0225i)29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.274 + 0.961i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.274 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57732 - 2.09017i\)
\(L(\frac12)\) \(\approx\) \(1.57732 - 2.09017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (-2.12 + 0.707i)T \)
41 \( 1 + (5 + 4i)T \)
good3 \( 1 + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.29 - 0.535i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + (-1.12 - 0.464i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (13.4 + 13.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (0.0502 + 0.121i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.07iT - 37T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (-0.636 - 1.53i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-1 - i)T + 61iT^{2} \)
67 \( 1 + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.24 - 4.24i)T - 73iT^{2} \)
79 \( 1 + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-5.87 - 14.1i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (5.46 - 13.1i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.00270991406158384242985371272, −9.496207578839313667600436623944, −8.594430516237340950408546382813, −7.06858255461615732545321871199, −6.31479347959714422514852267544, −5.40379897434951262225057729007, −4.54996891411066560441106768571, −3.48731035406610746280091347393, −2.24795517849300834216711997795, −1.13022358953124043980942546664, 1.98465420659321377532702778407, 3.08942416376126026417510023370, 4.39122206370372936587639496592, 5.25845284431046263266932409290, 6.02894126290954125291784266825, 7.01551329532964233072999673548, 7.61044420371063959752251121242, 8.676916614334566996754283119031, 9.637040185242249426742188588770, 10.43185891234634756004760391185

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