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Properties

Label 2-806-31.16-c1-0-20
Degree $2$
Conductor $806$
Sign $0.999 + 0.0385i$
Analytic cond. $6.43594$
Root an. cond. $2.53691$
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  + (0.809 − 0.587i)2-s + (2.64 + 1.91i)3-s + (0.309 − 0.951i)4-s − 0.940·5-s + 3.26·6-s + (0.903 − 2.77i)7-s + (−0.309 − 0.951i)8-s + (2.36 + 7.28i)9-s + (−0.760 + 0.552i)10-s + (1.41 − 4.36i)11-s + (2.64 − 1.91i)12-s + (0.809 + 0.587i)13-s + (−0.903 − 2.77i)14-s + (−2.48 − 1.80i)15-s + (−0.809 − 0.587i)16-s + (0.729 + 2.24i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (1.52 + 1.10i)3-s + (0.154 − 0.475i)4-s − 0.420·5-s + 1.33·6-s + (0.341 − 1.05i)7-s + (−0.109 − 0.336i)8-s + (0.788 + 2.42i)9-s + (−0.240 + 0.174i)10-s + (0.427 − 1.31i)11-s + (0.762 − 0.553i)12-s + (0.224 + 0.163i)13-s + (−0.241 − 0.742i)14-s + (−0.641 − 0.465i)15-s + (−0.202 − 0.146i)16-s + (0.176 + 0.544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(806\)    =    \(2 \cdot 13 \cdot 31\)
Sign: $0.999 + 0.0385i$
Analytic conductor: \(6.43594\)
Root analytic conductor: \(2.53691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{806} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 806,\ (\ :1/2),\ 0.999 + 0.0385i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.41419 - 0.0657665i\)
\(L(\frac12)\) \(\approx\) \(3.41419 - 0.0657665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (4.67 - 3.02i)T \)
good3 \( 1 + (-2.64 - 1.91i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 0.940T + 5T^{2} \)
7 \( 1 + (-0.903 + 2.77i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.41 + 4.36i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (-0.729 - 2.24i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.33 + 0.970i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.76 - 5.44i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.10 + 1.52i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 1.49T + 37T^{2} \)
41 \( 1 + (2.43 - 1.77i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (7.79 - 5.66i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (10.0 + 7.30i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.85 + 8.79i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.30 - 1.67i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 8.72T + 67T^{2} \)
71 \( 1 + (2.95 + 9.10i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.682 - 2.09i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.45 - 10.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (6.90 - 5.01i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (4.46 - 13.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-1.21 + 3.72i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.15435226385097790305039094040, −9.580528086230358022893840763177, −8.517041196232398360654596573838, −8.019138490315745259949144836615, −6.94157095644569165701953285954, −5.43091979129061009025444173956, −4.40894206708587226409174236398, −3.59609513380117978581036713454, −3.26763288750660766592305638397, −1.61209939676496433657543153249, 1.75824574083731546744671996392, 2.63828972712765562403214826125, 3.65442738954667239247103550529, 4.77141010754296362189650860728, 6.11025443416890229339235129272, 7.04447803661544890898991499617, 7.61480994454401413711352378337, 8.424117558187618878260879960656, 9.023818511033517830152227278932, 9.911843677173094272933465001181

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