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Properties

Label 2-816-12.11-c1-0-14
Degree $2$
Conductor $816$
Sign $0.595 - 0.803i$
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  + (1.39 + 1.03i)3-s − 2i·5-s + 2.78i·7-s + (0.870 + 2.87i)9-s − 3.50·11-s + 5.74·13-s + (2.06 − 2.78i)15-s − i·17-s + 5.56i·19-s + (−2.87 + 3.87i)21-s + 6.90·23-s + 25-s + (−1.75 + 4.89i)27-s + 5.48i·29-s + 3.50i·31-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)3-s − 0.894i·5-s + 1.05i·7-s + (0.290 + 0.956i)9-s − 1.05·11-s + 1.59·13-s + (0.532 − 0.718i)15-s − 0.242i·17-s + 1.27i·19-s + (−0.626 + 0.844i)21-s + 1.44·23-s + 0.200·25-s + (−0.336 + 0.941i)27-s + 1.01i·29-s + 0.628i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.595 - 0.803i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82735 + 0.919809i\)
\(L(\frac12)\) \(\approx\) \(1.82735 + 0.919809i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 - 1.03i)T \)
17 \( 1 + iT \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 - 2.78iT - 7T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 - 5.74T + 13T^{2} \)
19 \( 1 - 5.56iT - 19T^{2} \)
23 \( 1 - 6.90T + 23T^{2} \)
29 \( 1 - 5.48iT - 29T^{2} \)
31 \( 1 - 3.50iT - 31T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
41 \( 1 + 9.48iT - 41T^{2} \)
43 \( 1 + 7.00iT - 43T^{2} \)
47 \( 1 - 9.69T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 5.56T + 59T^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + 2.68T + 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 + 9.48T + 97T^{2} \)
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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

−10.45648476919701090864550463801, −9.134786663771348517239401774255, −8.754680370386410863466041710090, −8.260005833515474524588685506767, −7.07260483543256017810742406833, −5.48865225973390371568532002822, −5.23143897983701848277753772592, −3.86709826950398463167223486514, −2.93199922460405752022539587248, −1.61497892498532589684992301153, 1.04021602197524107612931584023, 2.64392060355830022889707227850, 3.37630699845029492009285667239, 4.47584001322086878388306497153, 6.01562599338496017207440988448, 6.90021192762411077200817250648, 7.43009521213713593059851370392, 8.333644782734034353529253510589, 9.128647930660012766352656387644, 10.27518267462592127586110437251

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