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Properties

Label 2-804-67.59-c1-0-3
Degree $2$
Conductor $804$
Sign $0.106 - 0.994i$
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  + (0.654 + 0.755i)3-s + (1.12 + 0.723i)5-s + (−0.202 + 1.40i)7-s + (−0.142 + 0.989i)9-s + (−2.52 − 1.62i)11-s + (1.08 + 2.37i)13-s + (0.190 + 1.32i)15-s + (2.15 − 0.633i)17-s + (1.08 + 7.55i)19-s + (−1.19 + 0.768i)21-s + (5.68 + 6.55i)23-s + (−1.33 − 2.91i)25-s + (−0.841 + 0.540i)27-s − 3.39·29-s + (1.49 − 3.26i)31-s + ⋯
L(s)  = 1  + (0.378 + 0.436i)3-s + (0.503 + 0.323i)5-s + (−0.0764 + 0.531i)7-s + (−0.0474 + 0.329i)9-s + (−0.762 − 0.489i)11-s + (0.301 + 0.659i)13-s + (0.0492 + 0.342i)15-s + (0.523 − 0.153i)17-s + (0.249 + 1.73i)19-s + (−0.260 + 0.167i)21-s + (1.18 + 1.36i)23-s + (−0.266 − 0.583i)25-s + (−0.161 + 0.104i)27-s − 0.630·29-s + (0.267 − 0.586i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.106 - 0.994i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.106 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31295 + 1.17973i\)
\(L(\frac12)\) \(\approx\) \(1.31295 + 1.17973i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (-0.105 - 8.18i)T \)
good5 \( 1 + (-1.12 - 0.723i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (0.202 - 1.40i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (2.52 + 1.62i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-1.08 - 2.37i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-2.15 + 0.633i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-1.08 - 7.55i)T + (-18.2 + 5.35i)T^{2} \)
23 \( 1 + (-5.68 - 6.55i)T + (-3.27 + 22.7i)T^{2} \)
29 \( 1 + 3.39T + 29T^{2} \)
31 \( 1 + (-1.49 + 3.26i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + 6.63T + 37T^{2} \)
41 \( 1 + (-1.86 + 0.547i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (8.87 - 2.60i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (0.660 + 0.762i)T + (-6.68 + 46.5i)T^{2} \)
53 \( 1 + (-11.0 - 3.23i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-3.66 + 8.03i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-8.80 + 5.65i)T + (25.3 - 55.4i)T^{2} \)
71 \( 1 + (3.96 + 1.16i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (6.59 - 4.23i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-3.97 - 8.70i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (7.38 + 4.74i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-7.60 + 8.77i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + 5.54T + 97T^{2} \)
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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

−10.20274632873674969922874595075, −9.757715410953550655124352479915, −8.792269474807225788369426730246, −8.045404395071557301644680046698, −7.05900850827338670240310514045, −5.81064320992407531195304741399, −5.36171521793147064292930864772, −3.88207723832412175763941385202, −2.99772276033889677795987096328, −1.79197795871362064844922286395, 0.869959620918854775967272435330, 2.34602634062586433731513626657, 3.37392140114564337320035478945, 4.79791943639518007172349619561, 5.52984484708173674226546450423, 6.85710456481349790772675074108, 7.33736663125283568892038395043, 8.471064992353700728851797105285, 9.070209220538378112514713270134, 10.15110470230975580350333210133

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