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Properties

Label 2-804-12.11-c1-0-5
Degree $2$
Conductor $804$
Sign $-0.335 - 0.942i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
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  + (−1.27 + 0.602i)2-s + (−1.48 − 0.887i)3-s + (1.27 − 1.54i)4-s − 3.40i·5-s + (2.43 + 0.239i)6-s + 4.07i·7-s + (−0.701 + 2.73i)8-s + (1.42 + 2.63i)9-s + (2.05 + 4.35i)10-s + 0.0181·11-s + (−3.26 + 1.16i)12-s − 1.41·13-s + (−2.45 − 5.20i)14-s + (−3.02 + 5.06i)15-s + (−0.752 − 3.92i)16-s − 2.82i·17-s + ⋯
L(s)  = 1  + (−0.904 + 0.425i)2-s + (−0.858 − 0.512i)3-s + (0.637 − 0.770i)4-s − 1.52i·5-s + (0.995 + 0.0976i)6-s + 1.53i·7-s + (−0.248 + 0.968i)8-s + (0.475 + 0.879i)9-s + (0.648 + 1.37i)10-s + 0.00548·11-s + (−0.942 + 0.335i)12-s − 0.392·13-s + (−0.655 − 1.39i)14-s + (−0.780 + 1.30i)15-s + (−0.188 − 0.982i)16-s − 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.335 - 0.942i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.335 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157343 + 0.223087i\)
\(L(\frac12)\) \(\approx\) \(0.157343 + 0.223087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.602i)T \)
3 \( 1 + (1.48 + 0.887i)T \)
67 \( 1 + iT \)
good5 \( 1 + 3.40iT - 5T^{2} \)
7 \( 1 - 4.07iT - 7T^{2} \)
11 \( 1 - 0.0181T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + 4.54iT - 19T^{2} \)
23 \( 1 + 8.65T + 23T^{2} \)
29 \( 1 - 9.67iT - 29T^{2} \)
31 \( 1 - 6.54iT - 31T^{2} \)
37 \( 1 + 5.85T + 37T^{2} \)
41 \( 1 + 1.44iT - 41T^{2} \)
43 \( 1 - 7.62iT - 43T^{2} \)
47 \( 1 - 0.189T + 47T^{2} \)
53 \( 1 - 13.6iT - 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 2.47T + 61T^{2} \)
71 \( 1 - 4.72T + 71T^{2} \)
73 \( 1 + 9.72T + 73T^{2} \)
79 \( 1 + 4.33iT - 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 13.7iT - 89T^{2} \)
97 \( 1 - 3.81T + 97T^{2} \)
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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

−10.40439426128535075250194038408, −9.353878192934398278805080105390, −8.838805253405577859707376231859, −8.128799600934091745280641138723, −7.11209527798763729667458987237, −6.12469007738705275048199826096, −5.25311406720837962084670986315, −4.92246727786454739525425394551, −2.39692556296551838771510494725, −1.29237658464451971733937058351, 0.21341044736715813631925127598, 2.05896334715870532918536746974, 3.74685641253861154022321636453, 3.95474890613861477355863932245, 5.96301747214030966146638898709, 6.65122620803823111877404306613, 7.42350893461689836571164317780, 8.126664663478507402923998296231, 9.875826768438657281649522918801, 10.09942076851411393986785054063

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