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Properties

Label 2-804-268.267-c1-0-2
Degree $2$
Conductor $804$
Sign $-0.304 - 0.952i$
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  + (−1.31 + 0.523i)2-s − 3-s + (1.45 − 1.37i)4-s − 1.71i·5-s + (1.31 − 0.523i)6-s − 3.43·7-s + (−1.18 + 2.56i)8-s + 9-s + (0.898 + 2.25i)10-s + 1.17·11-s + (−1.45 + 1.37i)12-s − 1.11i·13-s + (4.51 − 1.79i)14-s + 1.71i·15-s + (0.216 − 3.99i)16-s − 6.00·17-s + ⋯
L(s)  = 1  + (−0.928 + 0.370i)2-s − 0.577·3-s + (0.725 − 0.687i)4-s − 0.767i·5-s + (0.536 − 0.213i)6-s − 1.29·7-s + (−0.419 + 0.907i)8-s + 0.333·9-s + (0.284 + 0.713i)10-s + 0.352·11-s + (−0.419 + 0.397i)12-s − 0.308i·13-s + (1.20 − 0.480i)14-s + 0.443i·15-s + (0.0540 − 0.998i)16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.304 - 0.952i$
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: \((2,\ 804,\ (\ :1/2),\ -0.304 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185378 + 0.253882i\)
\(L(\frac12)\) \(\approx\) \(0.185378 + 0.253882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.523i)T \)
3 \( 1 + T \)
67 \( 1 + (3.55 - 7.37i)T \)
good5 \( 1 + 1.71iT - 5T^{2} \)
7 \( 1 + 3.43T + 7T^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 + 6.00T + 17T^{2} \)
19 \( 1 + 1.61iT - 19T^{2} \)
23 \( 1 - 5.58iT - 23T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 - 3.52T + 31T^{2} \)
37 \( 1 - 4.31T + 37T^{2} \)
41 \( 1 - 8.77iT - 41T^{2} \)
43 \( 1 - 1.55T + 43T^{2} \)
47 \( 1 - 3.12iT - 47T^{2} \)
53 \( 1 - 9.63iT - 53T^{2} \)
59 \( 1 + 5.48iT - 59T^{2} \)
61 \( 1 - 3.17iT - 61T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + 7.90T + 73T^{2} \)
79 \( 1 - 7.48T + 79T^{2} \)
83 \( 1 + 3.57iT - 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 7.37iT - 97T^{2} \)
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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

−10.34531262532359272580054406063, −9.404878864719264685910792771111, −9.119489810502045194885039123892, −8.007794996080641825864169238727, −6.94838959332406179423949501932, −6.34007973290312143878694925908, −5.48770484436373956539020977197, −4.35113867105677662016021713251, −2.77674510467960109352593040758, −1.12925062479673683430958988037, 0.25679119546566664696349043941, 2.17981309918463782942555863308, 3.24046882268778155194509097937, 4.30957703117386824142388136008, 6.11141769480818798395275720407, 6.64714339590753237520145075045, 7.20104640981079284127331474292, 8.538604514379387144971427405708, 9.306267979911623786379830362689, 10.09368703764850024040034226859

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