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Properties

Label 2-800-200.29-c1-0-26
Degree $2$
Conductor $800$
Sign $-0.591 - 0.806i$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
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  + (−2.03 − 1.48i)3-s + (−2.19 − 0.433i)5-s − 3.58i·7-s + (1.03 + 3.18i)9-s + (2.49 + 0.810i)11-s + (−2.11 − 6.50i)13-s + (3.82 + 4.13i)15-s + (−1.14 − 1.57i)17-s + (−3.00 − 4.14i)19-s + (−5.31 + 7.31i)21-s + (0.737 + 0.239i)23-s + (4.62 + 1.90i)25-s + (0.269 − 0.830i)27-s + (−3.28 + 4.52i)29-s + (−1.72 + 1.25i)31-s + ⋯
L(s)  = 1  + (−1.17 − 0.854i)3-s + (−0.981 − 0.193i)5-s − 1.35i·7-s + (0.344 + 1.06i)9-s + (0.752 + 0.244i)11-s + (−0.586 − 1.80i)13-s + (0.988 + 1.06i)15-s + (−0.277 − 0.381i)17-s + (−0.690 − 0.950i)19-s + (−1.15 + 1.59i)21-s + (0.153 + 0.0499i)23-s + (0.924 + 0.380i)25-s + (0.0519 − 0.159i)27-s + (−0.610 + 0.839i)29-s + (−0.309 + 0.224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.591 - 0.806i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148830 + 0.293743i\)
\(L(\frac12)\) \(\approx\) \(0.148830 + 0.293743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.19 + 0.433i)T \)
good3 \( 1 + (2.03 + 1.48i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 3.58iT - 7T^{2} \)
11 \( 1 + (-2.49 - 0.810i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.11 + 6.50i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.14 + 1.57i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (3.00 + 4.14i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.737 - 0.239i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.28 - 4.52i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.72 - 1.25i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.73 - 8.43i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.12 - 3.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.652T + 43T^{2} \)
47 \( 1 + (0.541 - 0.745i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.48 + 1.07i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.74 + 1.21i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.21 - 2.01i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.97 + 2.89i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.48 + 3.26i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (14.1 + 4.58i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.65 + 5.56i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.50 - 4.72i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.35 + 10.3i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.31 - 4.56i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.00928465894650991339272008970, −8.674362493538063475591154504895, −7.57103303360224160146797165639, −7.20995486746260780175167771940, −6.42374543481472945112252012924, −5.17781798582788857283543045127, −4.42309625420300578797693372528, −3.19355500507753076511892219871, −1.13048265352377051158750168476, −0.22455076793346967095053369633, 2.17697975809789282477781998763, 3.94901508465441424844493703264, 4.36046231968352451639570423667, 5.58798405744169571724293960266, 6.24437361832999446400694816320, 7.19328084793283673142999779201, 8.516085387969341106574119340174, 9.166311094867797608313865058698, 10.00927866888417480244025790455, 11.15354304936476466524906275239

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