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Properties

Label 2-816-17.4-c1-0-5
Degree $2$
Conductor $816$
Sign $0.907 - 0.420i$
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  + (0.707 − 0.707i)3-s + (−0.450 + 0.450i)5-s + (1.22 + 1.22i)7-s − 1.00i·9-s + (3.50 + 3.50i)11-s − 2.50·13-s + 0.636i·15-s + (2.17 + 3.50i)17-s + 0.950i·19-s + 1.72·21-s + (−1.50 − 1.50i)23-s + 4.59i·25-s + (−0.707 − 0.707i)27-s + (6.77 − 6.77i)29-s + (3.72 − 3.72i)31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.201 + 0.201i)5-s + (0.461 + 0.461i)7-s − 0.333i·9-s + (1.05 + 1.05i)11-s − 0.695·13-s + 0.164i·15-s + (0.528 + 0.849i)17-s + 0.218i·19-s + 0.377·21-s + (−0.312 − 0.312i)23-s + 0.918i·25-s + (−0.136 − 0.136i)27-s + (1.25 − 1.25i)29-s + (0.669 − 0.669i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.907 - 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80328 + 0.397731i\)
\(L(\frac12)\) \(\approx\) \(1.80328 + 0.397731i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-2.17 - 3.50i)T \)
good5 \( 1 + (0.450 - 0.450i)T - 5iT^{2} \)
7 \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \)
11 \( 1 + (-3.50 - 3.50i)T + 11iT^{2} \)
13 \( 1 + 2.50T + 13T^{2} \)
19 \( 1 - 0.950iT - 19T^{2} \)
23 \( 1 + (1.50 + 1.50i)T + 23iT^{2} \)
29 \( 1 + (-6.77 + 6.77i)T - 29iT^{2} \)
31 \( 1 + (-3.72 + 3.72i)T - 31iT^{2} \)
37 \( 1 + (6.00 - 6.00i)T - 37iT^{2} \)
41 \( 1 + (-4.89 - 4.89i)T + 41iT^{2} \)
43 \( 1 - 7.15iT - 43T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 + 6.44iT - 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 + (-2.09 - 2.09i)T + 61iT^{2} \)
67 \( 1 + 7.70T + 67T^{2} \)
71 \( 1 + (0.271 - 0.271i)T - 71iT^{2} \)
73 \( 1 + (-4.60 + 4.60i)T - 73iT^{2} \)
79 \( 1 + (1.67 + 1.67i)T + 79iT^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 - 1.98T + 89T^{2} \)
97 \( 1 + (-2.15 + 2.15i)T - 97iT^{2} \)
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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

−10.03645462799561041832080857365, −9.561073971487111365915120255945, −8.424502627814313304453301159310, −7.84336776885608924214230627574, −6.88457393969973200801917791883, −6.10454787241836819449902946390, −4.81478513261237432404537724830, −3.89964614911503571276269693089, −2.57685920146069715987458049866, −1.50131516098950595537578665258, 1.00793331181180064940047239585, 2.71507762367718925351404132423, 3.79628827284641426476410670252, 4.65858507581097523486093832551, 5.64258405221214830876080084682, 6.88149152909832185321866456793, 7.65191310960205352471809462451, 8.707025172787008271113653645930, 9.108917525876883819488576025071, 10.29065094204402369457165210032

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