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Properties

Label 2-804-201.164-c1-0-3
Degree $2$
Conductor $804$
Sign $0.349 - 0.937i$
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  + (−1.41 − 0.996i)3-s + 0.835·5-s + (2.58 + 1.49i)7-s + (1.01 + 2.82i)9-s + (−1.23 + 2.14i)11-s + (−2.02 + 1.16i)13-s + (−1.18 − 0.833i)15-s + (−5.34 + 3.08i)17-s + (0.255 + 0.441i)19-s + (−2.17 − 4.69i)21-s + (0.173 − 0.100i)23-s − 4.30·25-s + (1.38 − 5.00i)27-s + (0.0619 + 0.0357i)29-s + (9.16 + 5.28i)31-s + ⋯
L(s)  = 1  + (−0.817 − 0.575i)3-s + 0.373·5-s + (0.977 + 0.564i)7-s + (0.337 + 0.941i)9-s + (−0.373 + 0.647i)11-s + (−0.561 + 0.324i)13-s + (−0.305 − 0.215i)15-s + (−1.29 + 0.748i)17-s + (0.0585 + 0.101i)19-s + (−0.474 − 1.02i)21-s + (0.0361 − 0.0208i)23-s − 0.860·25-s + (0.265 − 0.964i)27-s + (0.0114 + 0.00663i)29-s + (1.64 + 0.949i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.349 - 0.937i$
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: \((2,\ 804,\ (\ :1/2),\ 0.349 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.857228 + 0.595496i\)
\(L(\frac12)\) \(\approx\) \(0.857228 + 0.595496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 0.996i)T \)
67 \( 1 + (3.66 + 7.32i)T \)
good5 \( 1 - 0.835T + 5T^{2} \)
7 \( 1 + (-2.58 - 1.49i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.23 - 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.02 - 1.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.34 - 3.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.255 - 0.441i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.173 + 0.100i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0619 - 0.0357i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.16 - 5.28i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.895 - 1.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.45 - 9.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 + (-1.53 - 0.886i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.08T + 53T^{2} \)
59 \( 1 - 2.34iT - 59T^{2} \)
61 \( 1 + (-2.52 + 1.45i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (-8.93 - 5.15i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.62 - 8.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.82 + 1.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.8 + 6.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.2iT - 89T^{2} \)
97 \( 1 + (10.8 - 6.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.48280046455037559045201312964, −9.759719190023881966666370531904, −8.522749218531725469498025656475, −7.87926590196744713135872310780, −6.82099747749433052593213766697, −6.11187489993172276342209330092, −5.02557332042580704430528545712, −4.51257427227968202347643927382, −2.42717797901068252925529907431, −1.60782474687527270154188763595, 0.57610818649069524058674429034, 2.33578336632547230760870701633, 3.90338376427740123119047091888, 4.80818198273872472873173939947, 5.48978673641444347206522571407, 6.51334604176237654964214148531, 7.45054763451127463298097818313, 8.449304242259636050421155444217, 9.427286828032851682608756590278, 10.25071846081219516941619104986

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