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Properties

Label 2-804-268.267-c1-0-58
Degree $2$
Conductor $804$
Sign $0.747 + 0.664i$
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  + (0.483 + 1.32i)2-s + 3-s + (−1.53 + 1.28i)4-s − 3.80i·5-s + (0.483 + 1.32i)6-s − 0.504·7-s + (−2.44 − 1.41i)8-s + 9-s + (5.04 − 1.83i)10-s − 0.340·11-s + (−1.53 + 1.28i)12-s − 4.62i·13-s + (−0.243 − 0.670i)14-s − 3.80i·15-s + (0.693 − 3.93i)16-s − 5.54·17-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + 0.577·3-s + (−0.765 + 0.642i)4-s − 1.69i·5-s + (0.197 + 0.542i)6-s − 0.190·7-s + (−0.866 − 0.499i)8-s + 0.333·9-s + (1.59 − 0.581i)10-s − 0.102·11-s + (−0.442 + 0.371i)12-s − 1.28i·13-s + (−0.0651 − 0.179i)14-s − 0.981i·15-s + (0.173 − 0.984i)16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.747 + 0.664i$
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: \((2,\ 804,\ (\ :1/2),\ 0.747 + 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51914 - 0.578081i\)
\(L(\frac12)\) \(\approx\) \(1.51914 - 0.578081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.483 - 1.32i)T \)
3 \( 1 - T \)
67 \( 1 + (1.18 + 8.09i)T \)
good5 \( 1 + 3.80iT - 5T^{2} \)
7 \( 1 + 0.504T + 7T^{2} \)
11 \( 1 + 0.340T + 11T^{2} \)
13 \( 1 + 4.62iT - 13T^{2} \)
17 \( 1 + 5.54T + 17T^{2} \)
19 \( 1 + 5.66iT - 19T^{2} \)
23 \( 1 - 2.26iT - 23T^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 - 3.48T + 31T^{2} \)
37 \( 1 - 3.22T + 37T^{2} \)
41 \( 1 + 5.82iT - 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 4.07iT - 47T^{2} \)
53 \( 1 - 2.30iT - 53T^{2} \)
59 \( 1 + 4.68iT - 59T^{2} \)
61 \( 1 - 5.81iT - 61T^{2} \)
71 \( 1 - 9.21iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 9.58T + 79T^{2} \)
83 \( 1 - 16.5iT - 83T^{2} \)
89 \( 1 + 9.37T + 89T^{2} \)
97 \( 1 - 0.449iT - 97T^{2} \)
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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

−9.614463384114256758339957605336, −9.103251137721111733104262022277, −8.381399694619407951283492636949, −7.79960184721748929472331020454, −6.70620893882084690869892539691, −5.58704562594270111055525661967, −4.82033950831482358042746773146, −4.09582087940148013484170992131, −2.70938838562646500015417750631, −0.67092269127545298165942966810, 1.98195679023498667102584438232, 2.73431771208077579197187244706, 3.72067679302569726613959296715, 4.52470804010632479463409677574, 6.19495294234353801599325512057, 6.66789453849654955104263959099, 7.86163874671412817503177052847, 8.920560801042506188287328260700, 9.765612339989259605410109210670, 10.41602911285352958488062576438

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