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Properties

Label 2-816-204.47-c1-0-4
Degree $2$
Conductor $816$
Sign $-0.588 - 0.808i$
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  + (−1.51 − 0.833i)3-s + (−1.19 + 1.19i)5-s + (3.01 + 3.01i)7-s + (1.60 + 2.53i)9-s + (2.35 − 2.35i)11-s − 6.82·13-s + (2.80 − 0.816i)15-s + (−2.80 − 3.02i)17-s + 1.36·19-s + (−2.06 − 7.09i)21-s + (−3.16 + 3.16i)23-s + 2.14i·25-s + (−0.330 − 5.18i)27-s + (−4.41 + 4.41i)29-s + (−1.81 + 1.81i)31-s + ⋯
L(s)  = 1  + (−0.876 − 0.481i)3-s + (−0.534 + 0.534i)5-s + (1.14 + 1.14i)7-s + (0.536 + 0.844i)9-s + (0.709 − 0.709i)11-s − 1.89·13-s + (0.725 − 0.210i)15-s + (−0.679 − 0.733i)17-s + 0.313·19-s + (−0.450 − 1.54i)21-s + (−0.660 + 0.660i)23-s + 0.429i·25-s + (−0.0636 − 0.997i)27-s + (−0.819 + 0.819i)29-s + (−0.325 + 0.325i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266209 + 0.523200i\)
\(L(\frac12)\) \(\approx\) \(0.266209 + 0.523200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 + 0.833i)T \)
17 \( 1 + (2.80 + 3.02i)T \)
good5 \( 1 + (1.19 - 1.19i)T - 5iT^{2} \)
7 \( 1 + (-3.01 - 3.01i)T + 7iT^{2} \)
11 \( 1 + (-2.35 + 2.35i)T - 11iT^{2} \)
13 \( 1 + 6.82T + 13T^{2} \)
19 \( 1 - 1.36T + 19T^{2} \)
23 \( 1 + (3.16 - 3.16i)T - 23iT^{2} \)
29 \( 1 + (4.41 - 4.41i)T - 29iT^{2} \)
31 \( 1 + (1.81 - 1.81i)T - 31iT^{2} \)
37 \( 1 + (-0.347 - 0.347i)T + 37iT^{2} \)
41 \( 1 + (-5.82 - 5.82i)T + 41iT^{2} \)
43 \( 1 - 4.83T + 43T^{2} \)
47 \( 1 + 7.97T + 47T^{2} \)
53 \( 1 + 7.31T + 53T^{2} \)
59 \( 1 - 13.5iT - 59T^{2} \)
61 \( 1 + (4.71 - 4.71i)T - 61iT^{2} \)
67 \( 1 - 2.73iT - 67T^{2} \)
71 \( 1 + (8.74 + 8.74i)T + 71iT^{2} \)
73 \( 1 + (1.93 + 1.93i)T + 73iT^{2} \)
79 \( 1 + (-3.96 - 3.96i)T + 79iT^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + 2.38iT - 89T^{2} \)
97 \( 1 + (12.6 + 12.6i)T + 97iT^{2} \)
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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

−10.94369399154524495787211934794, −9.672152025427091908507773610882, −8.837321055807441145312622737167, −7.61482865529341947166232921176, −7.31644101911069830001462460159, −6.09222729373042715955894048971, −5.26067257888084786033933263432, −4.52189496343775573597108017467, −2.86478479593982380421200773477, −1.69725299235864517126866880894, 0.32148114983790401995885358131, 1.87878446578577337076315559926, 4.13820010637950124811806960655, 4.36243561277925074406425265527, 5.19320558429616932075012518600, 6.52004481121944866392837628472, 7.41368915655966066749958993375, 8.035721730752878252805904114079, 9.355416859528696866475324172773, 9.992952874854405305240042830319

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