Properties

Label 2-804-268.267-c1-0-59
Degree $2$
Conductor $804$
Sign $0.364 + 0.931i$
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  + (1.35 − 0.418i)2-s + 3-s + (1.64 − 1.13i)4-s − 1.44i·5-s + (1.35 − 0.418i)6-s − 2.69·7-s + (1.75 − 2.21i)8-s + 9-s + (−0.606 − 1.95i)10-s + 3.96·11-s + (1.64 − 1.13i)12-s − 2.95i·13-s + (−3.64 + 1.13i)14-s − 1.44i·15-s + (1.44 − 3.73i)16-s − 2.16·17-s + ⋯
L(s)  = 1  + (0.955 − 0.296i)2-s + 0.577·3-s + (0.824 − 0.565i)4-s − 0.647i·5-s + (0.551 − 0.170i)6-s − 1.01·7-s + (0.620 − 0.784i)8-s + 0.333·9-s + (−0.191 − 0.618i)10-s + 1.19·11-s + (0.476 − 0.326i)12-s − 0.818i·13-s + (−0.974 + 0.302i)14-s − 0.374i·15-s + (0.360 − 0.932i)16-s − 0.524·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.364 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65221 - 1.80985i\)
\(L(\frac12)\) \(\approx\) \(2.65221 - 1.80985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.418i)T \)
3 \( 1 - T \)
67 \( 1 + (1.85 - 7.97i)T \)
good5 \( 1 + 1.44iT - 5T^{2} \)
7 \( 1 + 2.69T + 7T^{2} \)
11 \( 1 - 3.96T + 11T^{2} \)
13 \( 1 + 2.95iT - 13T^{2} \)
17 \( 1 + 2.16T + 17T^{2} \)
19 \( 1 - 6.52iT - 19T^{2} \)
23 \( 1 + 5.27iT - 23T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 + 4.41T + 31T^{2} \)
37 \( 1 + 6.67T + 37T^{2} \)
41 \( 1 + 0.555iT - 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 - 3.34iT - 53T^{2} \)
59 \( 1 - 7.97iT - 59T^{2} \)
61 \( 1 + 0.843iT - 61T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 + 8.46T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 8.13iT - 83T^{2} \)
89 \( 1 - 3.50T + 89T^{2} \)
97 \( 1 + 8.55iT - 97T^{2} \)
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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

−10.15587416083614081121412303531, −9.322655950205960555294248862725, −8.530719351240092430123800545572, −7.32847941561764995296948822341, −6.42858711078666648619807142077, −5.67464345035867085976265911320, −4.40039828046411802646869554749, −3.68350614047523364509051350886, −2.68361773736208323754420254863, −1.24023237265147984001456552647, 2.04800330262655361597712788800, 3.19707252018805760246493770712, 3.83138448385229060235607081992, 4.94240956059300168136337780331, 6.35516166286290243304087527801, 6.78068283119757461134462206489, 7.43096615874866402837123060170, 8.907928684104956130260968873428, 9.326339046824135470430657902662, 10.59163399386018502860478758247

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