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Properties

Label 2-804-268.267-c1-0-11
Degree $2$
Conductor $804$
Sign $-0.897 + 0.440i$
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  + (0.219 + 1.39i)2-s + 3-s + (−1.90 + 0.613i)4-s + 3.15i·5-s + (0.219 + 1.39i)6-s − 0.761·7-s + (−1.27 − 2.52i)8-s + 9-s + (−4.41 + 0.693i)10-s − 5.14·11-s + (−1.90 + 0.613i)12-s + 4.67i·13-s + (−0.167 − 1.06i)14-s + 3.15i·15-s + (3.24 − 2.33i)16-s + 1.78·17-s + ⋯
L(s)  = 1  + (0.155 + 0.987i)2-s + 0.577·3-s + (−0.951 + 0.306i)4-s + 1.41i·5-s + (0.0895 + 0.570i)6-s − 0.287·7-s + (−0.450 − 0.892i)8-s + 0.333·9-s + (−1.39 + 0.219i)10-s − 1.55·11-s + (−0.549 + 0.177i)12-s + 1.29i·13-s + (−0.0446 − 0.284i)14-s + 0.815i·15-s + (0.811 − 0.583i)16-s + 0.433·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.897 + 0.440i$
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: \((2,\ 804,\ (\ :1/2),\ -0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256869 - 1.10561i\)
\(L(\frac12)\) \(\approx\) \(0.256869 - 1.10561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.219 - 1.39i)T \)
3 \( 1 - T \)
67 \( 1 + (-8.09 + 1.18i)T \)
good5 \( 1 - 3.15iT - 5T^{2} \)
7 \( 1 + 0.761T + 7T^{2} \)
11 \( 1 + 5.14T + 11T^{2} \)
13 \( 1 - 4.67iT - 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 + 5.49iT - 19T^{2} \)
23 \( 1 - 2.96iT - 23T^{2} \)
29 \( 1 + 9.79T + 29T^{2} \)
31 \( 1 + 3.67T + 31T^{2} \)
37 \( 1 - 8.38T + 37T^{2} \)
41 \( 1 + 1.00iT - 41T^{2} \)
43 \( 1 - 7.22T + 43T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 - 9.25iT - 53T^{2} \)
59 \( 1 - 0.539iT - 59T^{2} \)
61 \( 1 - 2.35iT - 61T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + 7.50T + 73T^{2} \)
79 \( 1 + 4.41T + 79T^{2} \)
83 \( 1 - 8.49iT - 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 8.80iT - 97T^{2} \)
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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

−10.68090094601615020911128563205, −9.555756528641909725101229581949, −9.153934097919915305684603059723, −7.65743723006043201714324080833, −7.51174424475134423465695312631, −6.55845803883367752032304422039, −5.66385036992090371467039937775, −4.46511194653637359632015130745, −3.35792420495077057248132558097, −2.48084527821310198059812127123, 0.48368493825675038050398500421, 1.94694539948327406450079920177, 3.12304725353908224916732286718, 4.09548655535971380229451122930, 5.28305244507544865070486624023, 5.63583044043798073732404314849, 7.79440255391511891339869552462, 8.141276346189122199699394121040, 9.035093749957548652294382049147, 9.911866633922890045332654556330

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