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Properties

Label 2-804-12.11-c1-0-24
Degree $2$
Conductor $804$
Sign $0.995 - 0.0897i$
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  + (−1.40 − 0.109i)2-s + (−0.420 − 1.68i)3-s + (1.97 + 0.309i)4-s − 0.933i·5-s + (0.408 + 2.41i)6-s + 1.91i·7-s + (−2.75 − 0.653i)8-s + (−2.64 + 1.41i)9-s + (−0.102 + 1.31i)10-s − 1.29·11-s + (−0.311 − 3.45i)12-s + 3.83·13-s + (0.210 − 2.70i)14-s + (−1.56 + 0.392i)15-s + (3.80 + 1.22i)16-s + 7.71i·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0776i)2-s + (−0.242 − 0.970i)3-s + (0.987 + 0.154i)4-s − 0.417i·5-s + (0.166 + 0.985i)6-s + 0.723i·7-s + (−0.972 − 0.231i)8-s + (−0.882 + 0.471i)9-s + (−0.0324 + 0.416i)10-s − 0.390·11-s + (−0.0897 − 0.995i)12-s + 1.06·13-s + (0.0561 − 0.721i)14-s + (−0.404 + 0.101i)15-s + (0.952 + 0.305i)16-s + 1.87i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.995 - 0.0897i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.995 - 0.0897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808914 + 0.0363910i\)
\(L(\frac12)\) \(\approx\) \(0.808914 + 0.0363910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.109i)T \)
3 \( 1 + (0.420 + 1.68i)T \)
67 \( 1 + iT \)
good5 \( 1 + 0.933iT - 5T^{2} \)
7 \( 1 - 1.91iT - 7T^{2} \)
11 \( 1 + 1.29T + 11T^{2} \)
13 \( 1 - 3.83T + 13T^{2} \)
17 \( 1 - 7.71iT - 17T^{2} \)
19 \( 1 - 0.375iT - 19T^{2} \)
23 \( 1 - 1.08T + 23T^{2} \)
29 \( 1 + 3.74iT - 29T^{2} \)
31 \( 1 - 8.96iT - 31T^{2} \)
37 \( 1 + 7.86T + 37T^{2} \)
41 \( 1 - 7.73iT - 41T^{2} \)
43 \( 1 + 7.24iT - 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 5.24iT - 53T^{2} \)
59 \( 1 + 0.895T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 2.29iT - 89T^{2} \)
97 \( 1 + 2.62T + 97T^{2} \)
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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

−10.55445949464498630292800300033, −9.060112169659444392720036334196, −8.515869292874960502377360363782, −8.012466711170461746996140964251, −6.83483358493564187143331052763, −6.13926472118094819054985038240, −5.33168802969462249155648162381, −3.45700098665249445000160339486, −2.16150872486302994823583838543, −1.17480450384536733193466814099, 0.66975785341642831441905652230, 2.67341415113189689207108902484, 3.63250886973809829501555472289, 4.95285915189650686194010280536, 5.96825280843362565720171238093, 6.95302393711674590004002720150, 7.68810810261428288179471590863, 8.882093327821822435423315533731, 9.340347176119191265127781143646, 10.36991560335344125585434799852

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