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Properties

Label 2-816-51.5-c1-0-3
Degree $2$
Conductor $816$
Sign $-0.972 + 0.233i$
Analytic cond. $6.51579$
Root an. cond. $2.55260$
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  + (0.317 + 1.70i)3-s + (−2.04 − 0.407i)5-s + (0.635 + 3.19i)7-s + (−2.79 + 1.07i)9-s + (−2.09 + 3.13i)11-s + (0.507 + 0.507i)13-s + (0.0443 − 3.61i)15-s + (2.22 − 3.47i)17-s + (−2.68 − 6.48i)19-s + (−5.24 + 2.09i)21-s + (−1.28 − 0.857i)23-s + (−0.587 − 0.243i)25-s + (−2.72 − 4.42i)27-s + (−0.845 + 4.25i)29-s + (−1.88 + 1.25i)31-s + ⋯
L(s)  = 1  + (0.183 + 0.983i)3-s + (−0.916 − 0.182i)5-s + (0.240 + 1.20i)7-s + (−0.932 + 0.359i)9-s + (−0.631 + 0.944i)11-s + (0.140 + 0.140i)13-s + (0.0114 − 0.934i)15-s + (0.538 − 0.842i)17-s + (−0.616 − 1.48i)19-s + (−1.14 + 0.457i)21-s + (−0.267 − 0.178i)23-s + (−0.117 − 0.0486i)25-s + (−0.524 − 0.851i)27-s + (−0.157 + 0.789i)29-s + (−0.338 + 0.226i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-0.972 + 0.233i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ -0.972 + 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0722897 - 0.609682i\)
\(L(\frac12)\) \(\approx\) \(0.0722897 - 0.609682i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.317 - 1.70i)T \)
17 \( 1 + (-2.22 + 3.47i)T \)
good5 \( 1 + (2.04 + 0.407i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (-0.635 - 3.19i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (2.09 - 3.13i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-0.507 - 0.507i)T + 13iT^{2} \)
19 \( 1 + (2.68 + 6.48i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.28 + 0.857i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.845 - 4.25i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (1.88 - 1.25i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (1.30 + 1.95i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-6.01 + 1.19i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (4.24 - 10.2i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (9.31 - 9.31i)T - 47iT^{2} \)
53 \( 1 + (11.2 - 4.65i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.00377 - 0.00910i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.11 - 0.618i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + 9.67iT - 67T^{2} \)
71 \( 1 + (-5.55 + 3.70i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-0.349 + 1.75i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-8.98 - 6.00i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (1.92 + 4.64i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.15 - 8.15i)T + 89iT^{2} \)
97 \( 1 + (-3.22 - 0.640i)T + (89.6 + 37.1i)T^{2} \)
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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

−10.84707545245465323326040844366, −9.592100660035684448475414858424, −9.162650397005784126123473009471, −8.221193434871959643412785931630, −7.55099754025779711830804741468, −6.21276733617788227479350138051, −4.91778565303668971045699463741, −4.70935924426991489222155229148, −3.29221944633200676705527844239, −2.34826750833005291137700329444, 0.28668935872455690355013055122, 1.73194357330486307724137625187, 3.40324045145083087312463527806, 3.93616791582116868467189225156, 5.54614865765016454818642038256, 6.40772711832142072943933596599, 7.44495847714551834396781887244, 8.043213015698122660125876731257, 8.387995180041663018214882701220, 9.974917471149311311859817965539

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