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Properties

Label 2-804-12.11-c1-0-10
Degree $2$
Conductor $804$
Sign $-0.895 + 0.445i$
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  + (−0.0544 + 1.41i)2-s + (0.888 + 1.48i)3-s + (−1.99 − 0.153i)4-s − 4.13i·5-s + (−2.14 + 1.17i)6-s + 3.80i·7-s + (0.325 − 2.80i)8-s + (−1.41 + 2.64i)9-s + (5.84 + 0.225i)10-s − 2.63·11-s + (−1.54 − 3.10i)12-s − 3.22·13-s + (−5.37 − 0.207i)14-s + (6.14 − 3.67i)15-s + (3.95 + 0.613i)16-s + 4.45i·17-s + ⋯
L(s)  = 1  + (−0.0384 + 0.999i)2-s + (0.513 + 0.858i)3-s + (−0.997 − 0.0769i)4-s − 1.85i·5-s + (−0.877 + 0.479i)6-s + 1.43i·7-s + (0.115 − 0.993i)8-s + (−0.473 + 0.880i)9-s + (1.84 + 0.0712i)10-s − 0.793·11-s + (−0.445 − 0.895i)12-s − 0.893·13-s + (−1.43 − 0.0553i)14-s + (1.58 − 0.949i)15-s + (0.988 + 0.153i)16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.895 + 0.445i$
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: \((2,\ 804,\ (\ :1/2),\ -0.895 + 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180722 - 0.768617i\)
\(L(\frac12)\) \(\approx\) \(0.180722 - 0.768617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0544 - 1.41i)T \)
3 \( 1 + (-0.888 - 1.48i)T \)
67 \( 1 - iT \)
good5 \( 1 + 4.13iT - 5T^{2} \)
7 \( 1 - 3.80iT - 7T^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 + 3.22T + 13T^{2} \)
17 \( 1 - 4.45iT - 17T^{2} \)
19 \( 1 - 6.29iT - 19T^{2} \)
23 \( 1 + 2.53T + 23T^{2} \)
29 \( 1 + 0.152iT - 29T^{2} \)
31 \( 1 + 5.69iT - 31T^{2} \)
37 \( 1 + 6.40T + 37T^{2} \)
41 \( 1 - 2.22iT - 41T^{2} \)
43 \( 1 - 5.26iT - 43T^{2} \)
47 \( 1 + 0.0672T + 47T^{2} \)
53 \( 1 + 1.32iT - 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 - 14.7T + 61T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 - 6.91T + 73T^{2} \)
79 \( 1 - 7.73iT - 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 - 3.94iT - 89T^{2} \)
97 \( 1 + 2.17T + 97T^{2} \)
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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

−10.15351541301716375334597334376, −9.636084023027297109631261891743, −8.864609830391220363149044420645, −8.222754224228274343638745195428, −7.905437474348146008157018531290, −5.90128010631087297748945076883, −5.43930312258930236963283351159, −4.74042173985468470958255486619, −3.79977140920743106735293456958, −2.04078545752851758756358589899, 0.35326397300984165472782432419, 2.22791797027746981960266584286, 2.92266395443251275625482081885, 3.73769612059120592563849854865, 5.11385460997306382662837045280, 6.73955274955169749065717115613, 7.23378400400512666150262128319, 7.82404077161729101833584665335, 9.126841320427193941643374665567, 10.14380066638940993473750607754

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