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Properties

Label 2-804-67.40-c1-0-1
Degree $2$
Conductor $804$
Sign $-0.813 - 0.581i$
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 + (−0.960 − 0.281i)5-s + (0.00738 + 0.00852i)7-s + (−0.654 − 0.755i)9-s + (−0.0286 − 0.00841i)11-s + (−0.796 − 0.511i)13-s + (0.655 − 0.756i)15-s + (1.11 + 7.75i)17-s + (−0.720 + 0.831i)19-s + (−0.0108 + 0.00317i)21-s + (−2.46 + 5.38i)23-s + (−3.36 − 2.16i)25-s + (0.959 − 0.281i)27-s + 0.886·29-s + (−7.56 + 4.86i)31-s + ⋯
L(s)  = 1  + (−0.239 + 0.525i)3-s + (−0.429 − 0.126i)5-s + (0.00279 + 0.00322i)7-s + (−0.218 − 0.251i)9-s + (−0.00864 − 0.00253i)11-s + (−0.220 − 0.141i)13-s + (0.169 − 0.195i)15-s + (0.270 + 1.88i)17-s + (−0.165 + 0.190i)19-s + (−0.00236 + 0.000693i)21-s + (−0.513 + 1.12i)23-s + (−0.672 − 0.432i)25-s + (0.184 − 0.0542i)27-s + 0.164·29-s + (−1.35 + 0.873i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.813 - 0.581i$
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: \((2,\ 804,\ (\ :1/2),\ -0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200494 + 0.624712i\)
\(L(\frac12)\) \(\approx\) \(0.200494 + 0.624712i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.38 + 3.53i)T \)
good5 \( 1 + (0.960 + 0.281i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (-0.00738 - 0.00852i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (0.0286 + 0.00841i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.796 + 0.511i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-1.11 - 7.75i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.720 - 0.831i)T + (-2.70 - 18.8i)T^{2} \)
23 \( 1 + (2.46 - 5.38i)T + (-15.0 - 17.3i)T^{2} \)
29 \( 1 - 0.886T + 29T^{2} \)
31 \( 1 + (7.56 - 4.86i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + 6.19T + 37T^{2} \)
41 \( 1 + (0.919 + 6.39i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-0.139 - 0.971i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (2.18 - 4.77i)T + (-30.7 - 35.5i)T^{2} \)
53 \( 1 + (0.980 - 6.81i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (8.10 - 5.20i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-6.93 + 2.03i)T + (51.3 - 32.9i)T^{2} \)
71 \( 1 + (1.67 - 11.6i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (9.65 - 2.83i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-5.45 - 3.50i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-7.98 - 2.34i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (4.02 + 8.80i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−10.52534264019922814227736602698, −9.926126270212744242695822975275, −8.858048112520966979777295685797, −8.146622125029675079039589677246, −7.22526188088031830942995515880, −6.04733822148820649653268796548, −5.33816981540090061584459931301, −4.10199542904578692162653664714, −3.48472950669821277557470925781, −1.75938905125790124043585209088, 0.32550860963928045613677652073, 2.09189986641846956123537894969, 3.28765148953615613360230204850, 4.56595544350303769488910191506, 5.47589706970388995334481174171, 6.58571237374913563750696533716, 7.32856112110679819799626257949, 8.025414050668542115861406030603, 9.095017876210919425612593752145, 9.881250525324118361073406055131

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