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Properties

Label 2-804-201.38-c1-0-12
Degree $2$
Conductor $804$
Sign $0.860 + 0.509i$
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 − 1.41i)3-s + 2·5-s + (3.94 − 2.28i)7-s + (−1.00 + 2.82i)9-s + (1.72 + 2.98i)11-s + (4.5 + 2.59i)13-s + (−2 − 2.82i)15-s + (−2.17 − 1.25i)17-s + (−1.5 + 2.59i)19-s + (−7.17 − 3.30i)21-s + (8.17 + 4.71i)23-s − 25-s + (5.00 − 1.41i)27-s + (−7.62 + 4.40i)29-s + (−1.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + 0.894·5-s + (1.49 − 0.861i)7-s + (−0.333 + 0.942i)9-s + (0.520 + 0.900i)11-s + (1.24 + 0.720i)13-s + (−0.516 − 0.730i)15-s + (−0.527 − 0.304i)17-s + (−0.344 + 0.596i)19-s + (−1.56 − 0.721i)21-s + (1.70 + 0.984i)23-s − 0.200·25-s + (0.962 − 0.272i)27-s + (−1.41 + 0.817i)29-s + (−0.269 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77725 - 0.487261i\)
\(L(\frac12)\) \(\approx\) \(1.77725 - 0.487261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1 + 1.41i)T \)
67 \( 1 + (-8 + 1.73i)T \)
good5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (-3.94 + 2.28i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.72 - 2.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.17 + 1.25i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.17 - 4.71i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.62 - 4.40i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0505 - 0.0874i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.724 + 1.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 12.5iT - 43T^{2} \)
47 \( 1 + (-2.72 + 1.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 6.29iT - 59T^{2} \)
61 \( 1 + (0.398 + 0.230i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (3.82 - 2.20i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.94 - 3.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.39 - 5.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.07 + 4.08i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.0iT - 89T^{2} \)
97 \( 1 + (2.84 + 1.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.44485507188771630860668889400, −9.254903068134860823098196376774, −8.464577208197408834471004028553, −7.27812533951935677797552538244, −6.94823660127890426684070122216, −5.74522908135428033398181519164, −4.99437566785758875145643318145, −3.92633620664321168583923756926, −1.81472568817702691369000931991, −1.48650017155401316263308589237, 1.27838303588364949801333731156, 2.75597257200193589612248917687, 4.13133153782941967027965299294, 5.12293169248337983301333947536, 5.82625705482552160739370610173, 6.41664374478398514874201280560, 8.079189109646709803288926182825, 8.861719033509418689988245074992, 9.293149177623652420921538286867, 10.59983209227105559332709323987

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