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Properties

Label 2-804-67.40-c1-0-10
Degree $2$
Conductor $804$
Sign $-0.773 + 0.634i$
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  + (−0.415 + 0.909i)3-s + (1.61 + 0.472i)5-s + (−1.73 − 2.00i)7-s + (−0.654 − 0.755i)9-s + (−4.86 − 1.42i)11-s + (−1.52 − 0.981i)13-s + (−1.09 + 1.26i)15-s + (−0.741 − 5.15i)17-s + (−5.41 + 6.25i)19-s + (2.54 − 0.745i)21-s + (−1.76 + 3.86i)23-s + (−1.83 − 1.18i)25-s + (0.959 − 0.281i)27-s − 5.92·29-s + (5.48 − 3.52i)31-s + ⋯
L(s)  = 1  + (−0.239 + 0.525i)3-s + (0.720 + 0.211i)5-s + (−0.655 − 0.756i)7-s + (−0.218 − 0.251i)9-s + (−1.46 − 0.431i)11-s + (−0.423 − 0.272i)13-s + (−0.283 + 0.327i)15-s + (−0.179 − 1.25i)17-s + (−1.24 + 1.43i)19-s + (0.554 − 0.162i)21-s + (−0.368 + 0.806i)23-s + (−0.367 − 0.236i)25-s + (0.184 − 0.0542i)27-s − 1.10·29-s + (0.985 − 0.633i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.773 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0996381 - 0.278675i\)
\(L(\frac12)\) \(\approx\) \(0.0996381 - 0.278675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (1.49 + 8.04i)T \)
good5 \( 1 + (-1.61 - 0.472i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (1.73 + 2.00i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (4.86 + 1.42i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (1.52 + 0.981i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.741 + 5.15i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (5.41 - 6.25i)T + (-2.70 - 18.8i)T^{2} \)
23 \( 1 + (1.76 - 3.86i)T + (-15.0 - 17.3i)T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 + (-5.48 + 3.52i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 - 3.97T + 37T^{2} \)
41 \( 1 + (0.509 + 3.54i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-0.150 - 1.04i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (2.23 - 4.88i)T + (-30.7 - 35.5i)T^{2} \)
53 \( 1 + (0.275 - 1.91i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (0.666 - 0.428i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-0.419 + 0.123i)T + (51.3 - 32.9i)T^{2} \)
71 \( 1 + (-0.458 + 3.19i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-6.68 + 1.96i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (9.05 + 5.81i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (16.4 + 4.83i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-4.61 - 10.1i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−10.01485716542877993546370239010, −9.455509892203782608008288675106, −8.128356421624056364905819085312, −7.40297714627346457582896876462, −6.18772943383704058359587830556, −5.61413320625874060081341205739, −4.50803199050309579322123585554, −3.37929053668421227304707645395, −2.29348870317435567537807945043, −0.13480060327246327748353031125, 2.03955227423843653113971786768, 2.68767067929222042774917669084, 4.46264242925322252669045131076, 5.45331481808172566604398573754, 6.20042736636850624950151986248, 6.97795841098932599359438769165, 8.133878734688938809406824810448, 8.840307029555487968806148000982, 9.821945500787074776365560575232, 10.48297771639327517048646826922

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