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Properties

Label 2-804-67.25-c1-0-3
Degree $2$
Conductor $804$
Sign $0.0446 - 0.999i$
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 + (−0.494 + 0.317i)5-s + (0.597 + 4.15i)7-s + (−0.142 − 0.989i)9-s + (−3.69 + 2.37i)11-s + (0.305 − 0.668i)13-s + (−0.0836 + 0.581i)15-s + (1.69 + 0.498i)17-s + (−0.290 + 2.02i)19-s + (3.52 + 2.26i)21-s + (−4.46 + 5.15i)23-s + (−1.93 + 4.23i)25-s + (−0.841 − 0.540i)27-s − 4.58·29-s + (−1.01 − 2.21i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (−0.221 + 0.142i)5-s + (0.225 + 1.56i)7-s + (−0.0474 − 0.329i)9-s + (−1.11 + 0.715i)11-s + (0.0846 − 0.185i)13-s + (−0.0216 + 0.150i)15-s + (0.411 + 0.120i)17-s + (−0.0666 + 0.463i)19-s + (0.770 + 0.495i)21-s + (−0.930 + 1.07i)23-s + (−0.386 + 0.846i)25-s + (−0.161 − 0.104i)27-s − 0.850·29-s + (−0.181 − 0.398i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0446 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0446 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.0446 - 0.999i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.0446 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934095 + 0.893291i\)
\(L(\frac12)\) \(\approx\) \(0.934095 + 0.893291i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 - 8.07i)T \)
good5 \( 1 + (0.494 - 0.317i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.597 - 4.15i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (3.69 - 2.37i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.305 + 0.668i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-1.69 - 0.498i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (0.290 - 2.02i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (4.46 - 5.15i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 + (1.01 + 2.21i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 - 5.57T + 37T^{2} \)
41 \( 1 + (3.79 + 1.11i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-9.78 - 2.87i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-4.93 + 5.68i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (-7.24 + 2.12i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-5.10 - 11.1i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-8.08 - 5.19i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (-0.981 + 0.288i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (8.46 + 5.44i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-2.34 + 5.14i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-3.95 + 2.54i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (2.92 + 3.37i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 - 8.38T + 97T^{2} \)
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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

−10.35046850046785287271717476484, −9.525474658284424758191397490113, −8.710879589166772635533745608313, −7.82312692453284033919111819353, −7.34072968242052116548744919265, −5.76814856386623844952465130435, −5.51065248351189509603915622013, −3.95898763303677539818610087346, −2.70785159186715060203759979118, −1.89135955843039694116711721738, 0.59603186959352760961673687966, 2.47120620508142025841485791423, 3.75324097868258187177211085705, 4.40354385821692023145227967591, 5.47829784662061398680059924785, 6.69028862198372168198856338596, 7.77888368736991943452755414076, 8.115390363114698427750104948003, 9.253708780430196981913002273845, 10.28090939858209821081964550214

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