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Properties

Label 2-804-201.110-c1-0-10
Degree $2$
Conductor $804$
Sign $0.227 - 0.973i$
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  + (0.936 + 1.45i)3-s + (4.81 + 2.19i)7-s + (−1.24 + 2.72i)9-s + (−0.298 + 1.01i)13-s + (−2.01 − 4.41i)19-s + (1.30 + 9.07i)21-s + (4.79 + 1.40i)25-s + (−5.14 + 0.739i)27-s + (−1.53 − 5.21i)31-s − 7.64·37-s + (−1.75 + 0.516i)39-s + (5.83 + 5.05i)43-s + (13.7 + 15.8i)49-s + (4.54 − 7.07i)57-s + (1.72 − 0.247i)61-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)3-s + (1.81 + 0.830i)7-s + (−0.415 + 0.909i)9-s + (−0.0827 + 0.281i)13-s + (−0.462 − 1.01i)19-s + (0.284 + 1.97i)21-s + (0.959 + 0.281i)25-s + (−0.989 + 0.142i)27-s + (−0.275 − 0.937i)31-s − 1.25·37-s + (−0.281 + 0.0827i)39-s + (0.889 + 0.770i)43-s + (1.96 + 2.26i)49-s + (0.602 − 0.937i)57-s + (0.220 − 0.0317i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.227 - 0.973i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (713, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68721 + 1.33844i\)
\(L(\frac12)\) \(\approx\) \(1.68721 + 1.33844i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.936 - 1.45i)T \)
67 \( 1 + (8.16 + 0.591i)T \)
good5 \( 1 + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-4.81 - 2.19i)T + (4.58 + 5.29i)T^{2} \)
11 \( 1 + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.298 - 1.01i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (2.41 + 16.8i)T^{2} \)
19 \( 1 + (2.01 + 4.41i)T + (-12.4 + 14.3i)T^{2} \)
23 \( 1 + (-9.55 + 20.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.53 + 5.21i)T + (-26.0 + 16.7i)T^{2} \)
37 \( 1 + 7.64T + 37T^{2} \)
41 \( 1 + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-5.83 - 5.05i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + (-19.5 + 42.7i)T^{2} \)
53 \( 1 + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-1.72 + 0.247i)T + (58.5 - 17.1i)T^{2} \)
71 \( 1 + (10.1 - 70.2i)T^{2} \)
73 \( 1 + (1.74 + 12.1i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (4.36 - 14.8i)T + (-66.4 - 42.7i)T^{2} \)
83 \( 1 + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-36.9 - 80.9i)T^{2} \)
97 \( 1 + 19.2iT - 97T^{2} \)
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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

−10.59859486470338050288666630682, −9.353354032301703426160622877985, −8.785374038007723895245519693619, −8.143325429791524527839422861502, −7.23170829517441776652969949160, −5.73379683375644523216148262053, −4.89964204310630340610801581701, −4.30825724939015439339850304991, −2.82165545247707761898046785724, −1.84287196062897168595852298711, 1.14443104812082336923551578578, 2.09156115006110979873075139247, 3.55739008509824932105025603298, 4.62488888230319745489050312619, 5.65449769196373938542769834015, 6.92603743868490162681160215886, 7.54713372206303708353934693704, 8.330618005004032445965012720305, 8.835292324167611368022974427288, 10.29560170040771338049410067470

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