Properties

Label 2-804-268.267-c1-0-64
Degree $2$
Conductor $804$
Sign $-0.524 + 0.851i$
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  + (0.946 − 1.05i)2-s + 3-s + (−0.208 − 1.98i)4-s − 4.42i·5-s + (0.946 − 1.05i)6-s + 2.81·7-s + (−2.28 − 1.66i)8-s + 9-s + (−4.65 − 4.19i)10-s + 4.49·11-s + (−0.208 − 1.98i)12-s + 6.44i·13-s + (2.66 − 2.96i)14-s − 4.42i·15-s + (−3.91 + 0.831i)16-s + 1.33·17-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + 0.577·3-s + (−0.104 − 0.994i)4-s − 1.98i·5-s + (0.386 − 0.429i)6-s + 1.06·7-s + (−0.808 − 0.587i)8-s + 0.333·9-s + (−1.47 − 1.32i)10-s + 1.35·11-s + (−0.0603 − 0.574i)12-s + 1.78i·13-s + (0.712 − 0.791i)14-s − 1.14i·15-s + (−0.978 + 0.207i)16-s + 0.323·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.524 + 0.851i$
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: \((2,\ 804,\ (\ :1/2),\ -0.524 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45500 - 2.60437i\)
\(L(\frac12)\) \(\approx\) \(1.45500 - 2.60437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.946 + 1.05i)T \)
3 \( 1 - T \)
67 \( 1 + (6.48 + 4.99i)T \)
good5 \( 1 + 4.42iT - 5T^{2} \)
7 \( 1 - 2.81T + 7T^{2} \)
11 \( 1 - 4.49T + 11T^{2} \)
13 \( 1 - 6.44iT - 13T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 - 1.70iT - 19T^{2} \)
23 \( 1 - 3.41iT - 23T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 + 8.45T + 31T^{2} \)
37 \( 1 - 5.75T + 37T^{2} \)
41 \( 1 - 5.15iT - 41T^{2} \)
43 \( 1 + 1.06T + 43T^{2} \)
47 \( 1 - 4.52iT - 47T^{2} \)
53 \( 1 - 4.63iT - 53T^{2} \)
59 \( 1 + 10.2iT - 59T^{2} \)
61 \( 1 - 9.66iT - 61T^{2} \)
71 \( 1 + 0.479iT - 71T^{2} \)
73 \( 1 - 3.68T + 73T^{2} \)
79 \( 1 - 9.91T + 79T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−9.547579020418817213975991735242, −9.309783335858710618732331760369, −8.629817373884982389913018301986, −7.56413391461962434274013967540, −6.16135023402138816310315312519, −5.12668193266789644319023167626, −4.33833296110290089539309500263, −3.86822167803576465186280195162, −1.70138153561820949620897959108, −1.47307042151190013346026417793, 2.28008976829764549839016074722, 3.29245408312173660757933625689, 3.95985539681378689489408392010, 5.37814419726990218860082220050, 6.28529438596998242806140119160, 7.17754597117143209437493947425, 7.68778022062201706783587906632, 8.516340372453144480775662496835, 9.663910477523645655820984286649, 10.75583204730280342740873743470

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