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Properties

Label 2-804-201.41-c1-0-1
Degree $2$
Conductor $804$
Sign $-0.748 - 0.663i$
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  + (0.354 − 1.69i)3-s + (−0.606 + 4.21i)5-s + (−0.295 − 3.08i)7-s + (−2.74 − 1.20i)9-s + (−5.34 + 2.14i)11-s + (1.06 + 1.11i)13-s + (6.93 + 2.52i)15-s + (−3.13 + 1.08i)17-s + (0.373 + 0.0356i)19-s + (−5.34 − 0.596i)21-s + (3.36 − 0.160i)23-s + (−12.6 − 3.70i)25-s + (−3.01 + 4.23i)27-s + (−5.55 + 3.20i)29-s + (−5.06 + 5.31i)31-s + ⋯
L(s)  = 1  + (0.204 − 0.978i)3-s + (−0.271 + 1.88i)5-s + (−0.111 − 1.16i)7-s + (−0.916 − 0.401i)9-s + (−1.61 + 0.645i)11-s + (0.294 + 0.309i)13-s + (1.79 + 0.651i)15-s + (−0.760 + 0.263i)17-s + (0.0856 + 0.00817i)19-s + (−1.16 − 0.130i)21-s + (0.700 − 0.0333i)23-s + (−2.52 − 0.740i)25-s + (−0.580 + 0.814i)27-s + (−1.03 + 0.595i)29-s + (−0.909 + 0.954i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.748 - 0.663i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.748 - 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124038 + 0.326985i\)
\(L(\frac12)\) \(\approx\) \(0.124038 + 0.326985i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.354 + 1.69i)T \)
67 \( 1 + (-1.49 - 8.04i)T \)
good5 \( 1 + (0.606 - 4.21i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.295 + 3.08i)T + (-6.87 + 1.32i)T^{2} \)
11 \( 1 + (5.34 - 2.14i)T + (7.96 - 7.59i)T^{2} \)
13 \( 1 + (-1.06 - 1.11i)T + (-0.618 + 12.9i)T^{2} \)
17 \( 1 + (3.13 - 1.08i)T + (13.3 - 10.5i)T^{2} \)
19 \( 1 + (-0.373 - 0.0356i)T + (18.6 + 3.59i)T^{2} \)
23 \( 1 + (-3.36 + 0.160i)T + (22.8 - 2.18i)T^{2} \)
29 \( 1 + (5.55 - 3.20i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.06 - 5.31i)T + (-1.47 - 30.9i)T^{2} \)
37 \( 1 + (-2.92 + 5.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.4 + 2.00i)T + (38.0 + 15.2i)T^{2} \)
43 \( 1 + (-0.118 - 0.102i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + (-2.91 + 5.64i)T + (-27.2 - 38.2i)T^{2} \)
53 \( 1 + (-1.39 - 1.61i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-0.864 - 2.94i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (5.18 - 12.9i)T + (-44.1 - 42.0i)T^{2} \)
71 \( 1 + (-4.54 - 1.57i)T + (55.8 + 43.8i)T^{2} \)
73 \( 1 + (-2.76 - 1.10i)T + (52.8 + 50.3i)T^{2} \)
79 \( 1 + (-0.363 + 0.0882i)T + (70.2 - 36.1i)T^{2} \)
83 \( 1 + (-6.94 + 8.82i)T + (-19.5 - 80.6i)T^{2} \)
89 \( 1 + (1.23 - 1.92i)T + (-36.9 - 80.9i)T^{2} \)
97 \( 1 + (0.393 + 0.227i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.78632061004723296577913194552, −10.05058286843003671375518007332, −8.696325408684853333438892108607, −7.50245385030984728174519301934, −7.25388703072140840808309982040, −6.68510342396774154466823846598, −5.49510653870845793155920366405, −3.89254444688891394743850848808, −2.99101520933009623343742694493, −2.02362188333606177196579004040, 0.15456173909010658323395582384, 2.27098739894883819172430605856, 3.49764804861612514746625948508, 4.76700911608422377046841289155, 5.26062597157144491362279063133, 5.90317536739071958518755225467, 7.998015561369883805803218239492, 8.267747318799083924579784754271, 9.204751813257204107298690510469, 9.523392061813264821301957432084

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