Properties

Label 2-804-67.54-c1-0-3
Degree $2$
Conductor $804$
Sign $0.882 - 0.470i$
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 + (−2.07 + 1.33i)5-s + (4.33 + 1.73i)7-s + (−0.142 − 0.989i)9-s + (−0.101 − 2.12i)11-s + (−3.24 + 0.309i)13-s + (−0.350 + 2.43i)15-s + (1.86 + 7.70i)17-s + (6.34 − 2.53i)19-s + (4.14 − 2.13i)21-s + (1.25 + 0.241i)23-s + (0.446 − 0.978i)25-s + (−0.841 − 0.540i)27-s + (−3.89 + 6.75i)29-s + (6.64 + 0.634i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (−0.927 + 0.595i)5-s + (1.63 + 0.655i)7-s + (−0.0474 − 0.329i)9-s + (−0.0304 − 0.640i)11-s + (−0.899 + 0.0859i)13-s + (−0.0905 + 0.629i)15-s + (0.453 + 1.86i)17-s + (1.45 − 0.582i)19-s + (0.905 − 0.466i)21-s + (0.261 + 0.0503i)23-s + (0.0893 − 0.195i)25-s + (−0.161 − 0.104i)27-s + (−0.724 + 1.25i)29-s + (1.19 + 0.113i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69077 + 0.422200i\)
\(L(\frac12)\) \(\approx\) \(1.69077 + 0.422200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.88 + 7.20i)T \)
good5 \( 1 + (2.07 - 1.33i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-4.33 - 1.73i)T + (5.06 + 4.83i)T^{2} \)
11 \( 1 + (0.101 + 2.12i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (3.24 - 0.309i)T + (12.7 - 2.46i)T^{2} \)
17 \( 1 + (-1.86 - 7.70i)T + (-15.1 + 7.78i)T^{2} \)
19 \( 1 + (-6.34 + 2.53i)T + (13.7 - 13.1i)T^{2} \)
23 \( 1 + (-1.25 - 0.241i)T + (21.3 + 8.54i)T^{2} \)
29 \( 1 + (3.89 - 6.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.64 - 0.634i)T + (30.4 + 5.86i)T^{2} \)
37 \( 1 + (1.01 + 1.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.72 + 6.41i)T + (1.95 - 40.9i)T^{2} \)
43 \( 1 + (-2.12 - 0.624i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-0.0968 - 0.279i)T + (-36.9 + 29.0i)T^{2} \)
53 \( 1 + (2.36 - 0.695i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-3.15 - 6.91i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (0.399 - 8.38i)T + (-60.7 - 5.79i)T^{2} \)
71 \( 1 + (-2.12 + 8.75i)T + (-63.1 - 32.5i)T^{2} \)
73 \( 1 + (-0.736 + 15.4i)T + (-72.6 - 6.93i)T^{2} \)
79 \( 1 + (-6.90 - 9.69i)T + (-25.8 + 74.6i)T^{2} \)
83 \( 1 + (0.285 + 0.147i)T + (48.1 + 67.6i)T^{2} \)
89 \( 1 + (7.78 + 8.98i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-3.21 - 5.57i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.63638409213523987255401015950, −9.226837599386970290790227848715, −8.436306549026423618402339967748, −7.72932080269744800038301250578, −7.26387921461347187944057115441, −5.88029780011754735449591056900, −4.98075233716905745030473189411, −3.78558528742788246705365647968, −2.73431457640766110964674277672, −1.43632976007169300157970646254, 0.982606883002684275648291757144, 2.59731105744456318441588427658, 4.02018183551169010821070208877, 4.76022297260163921160526139803, 5.24490788804920358642726758170, 7.23842225178996121711445712860, 7.74752386145284717976339663307, 8.208590812679539505214932608561, 9.532399800765759899782292495621, 9.942454288027918840360669662118

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