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Properties

Label 2-804-201.164-c1-0-0
Degree $2$
Conductor $804$
Sign $-0.728 - 0.684i$
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  + (1.72 − 0.188i)3-s − 3.09·5-s + (−1.38 − 0.800i)7-s + (2.92 − 0.650i)9-s + (−2.72 + 4.71i)11-s + (−3.69 + 2.13i)13-s + (−5.32 + 0.584i)15-s + (−1.10 + 0.638i)17-s + (0.117 + 0.204i)19-s + (−2.53 − 1.11i)21-s + (−4.09 + 2.36i)23-s + 4.57·25-s + (4.91 − 1.67i)27-s + (1.17 + 0.676i)29-s + (−1.21 − 0.701i)31-s + ⋯
L(s)  = 1  + (0.994 − 0.109i)3-s − 1.38·5-s + (−0.523 − 0.302i)7-s + (0.976 − 0.216i)9-s + (−0.821 + 1.42i)11-s + (−1.02 + 0.591i)13-s + (−1.37 + 0.150i)15-s + (−0.268 + 0.154i)17-s + (0.0270 + 0.0468i)19-s + (−0.553 − 0.243i)21-s + (−0.853 + 0.492i)23-s + 0.914·25-s + (0.946 − 0.322i)27-s + (0.217 + 0.125i)29-s + (−0.218 − 0.126i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.728 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216576 + 0.546718i\)
\(L(\frac12)\) \(\approx\) \(0.216576 + 0.546718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.188i)T \)
67 \( 1 + (-6.48 - 4.99i)T \)
good5 \( 1 + 3.09T + 5T^{2} \)
7 \( 1 + (1.38 + 0.800i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.72 - 4.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.69 - 2.13i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.10 - 0.638i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.117 - 0.204i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.09 - 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.17 - 0.676i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.21 + 0.701i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.33 - 5.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.91 - 6.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 7.91iT - 43T^{2} \)
47 \( 1 + (0.523 + 0.302i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.20T + 53T^{2} \)
59 \( 1 - 13.8iT - 59T^{2} \)
61 \( 1 + (-7.23 + 4.17i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (11.3 + 6.53i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.23 + 5.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.34 + 4.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.5 + 7.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.12iT - 89T^{2} \)
97 \( 1 + (-2.75 + 1.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.21170696178837331784546492483, −9.851144461991094816925951108782, −8.797262371884356781652290281078, −7.78571107767405418875032566062, −7.45133830564157250254102251664, −6.65707266615783091629939153551, −4.80267533820635376968741559675, −4.17954910084802161413519757073, −3.17296768019085817172045327412, −2.01977677329909582302751371059, 0.24225729991679796467275167182, 2.58258295602160589205367104820, 3.29577804221470614116329147177, 4.22125215919681107068119249693, 5.35364452793532478014545532875, 6.65795894584717240261935214185, 7.81142293837331150951400524397, 8.027915764960129018067806090511, 8.933450778567627158801101377070, 9.869635087475603242460809909469

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