Properties

Label 2-804-268.267-c1-0-18
Degree $2$
Conductor $804$
Sign $0.681 - 0.731i$
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  + (−0.271 + 1.38i)2-s − 3-s + (−1.85 − 0.754i)4-s − 0.299i·5-s + (0.271 − 1.38i)6-s − 1.88·7-s + (1.55 − 2.36i)8-s + 9-s + (0.416 + 0.0814i)10-s − 2.47·11-s + (1.85 + 0.754i)12-s + 1.99i·13-s + (0.511 − 2.60i)14-s + 0.299i·15-s + (2.86 + 2.79i)16-s + 0.452·17-s + ⋯
L(s)  = 1  + (−0.192 + 0.981i)2-s − 0.577·3-s + (−0.926 − 0.377i)4-s − 0.134i·5-s + (0.110 − 0.566i)6-s − 0.710·7-s + (0.548 − 0.836i)8-s + 0.333·9-s + (0.131 + 0.0257i)10-s − 0.744·11-s + (0.534 + 0.217i)12-s + 0.554i·13-s + (0.136 − 0.697i)14-s + 0.0774i·15-s + (0.715 + 0.698i)16-s + 0.109·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.681 - 0.731i$
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: \((2,\ 804,\ (\ :1/2),\ 0.681 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783759 + 0.341012i\)
\(L(\frac12)\) \(\approx\) \(0.783759 + 0.341012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.271 - 1.38i)T \)
3 \( 1 + T \)
67 \( 1 + (-2.90 + 7.65i)T \)
good5 \( 1 + 0.299iT - 5T^{2} \)
7 \( 1 + 1.88T + 7T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 - 1.99iT - 13T^{2} \)
17 \( 1 - 0.452T + 17T^{2} \)
19 \( 1 + 5.21iT - 19T^{2} \)
23 \( 1 + 0.807iT - 23T^{2} \)
29 \( 1 - 9.00T + 29T^{2} \)
31 \( 1 - 8.22T + 31T^{2} \)
37 \( 1 - 1.31T + 37T^{2} \)
41 \( 1 - 6.67iT - 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 - 2.51iT - 47T^{2} \)
53 \( 1 - 0.990iT - 53T^{2} \)
59 \( 1 - 7.56iT - 59T^{2} \)
61 \( 1 + 9.06iT - 61T^{2} \)
71 \( 1 - 0.0862iT - 71T^{2} \)
73 \( 1 - 1.91T + 73T^{2} \)
79 \( 1 - 8.30T + 79T^{2} \)
83 \( 1 - 0.827iT - 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 6.11iT - 97T^{2} \)
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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

−10.19815618939584894482870846756, −9.468102867119053866799919557516, −8.610443753410556879657184006552, −7.71155115023407363216829347748, −6.63905845223959442734830527149, −6.31977029991516718814352079150, −5.02090054263504263254343674784, −4.50915847272661732419653250056, −2.91029007033710530096125756915, −0.75672055331039944556574291531, 0.852951875550104829337650171678, 2.53747671886258372613739419937, 3.47030329818192676542172698890, 4.65361681016110541434554051178, 5.57279042470416873372512488588, 6.59800255191303480459350451614, 7.79874913021461494490420495091, 8.529144455603966060671568390228, 9.684014187494583488581721738656, 10.32943373538909722333585428206

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