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Properties

Label 2-816-12.11-c1-0-31
Degree $2$
Conductor $816$
Sign $-0.836 - 0.547i$
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  + (−0.781 − 1.54i)3-s − 1.94i·5-s − 3.97i·7-s + (−1.77 + 2.41i)9-s − 5.57·11-s − 1.25·13-s + (−3.00 + 1.51i)15-s + i·17-s + 2.07i·19-s + (−6.13 + 3.10i)21-s + 4.52·23-s + 1.22·25-s + (5.12 + 0.861i)27-s + 7.39i·29-s − 8.94i·31-s + ⋯
L(s)  = 1  + (−0.451 − 0.892i)3-s − 0.868i·5-s − 1.50i·7-s + (−0.592 + 0.805i)9-s − 1.68·11-s − 0.347·13-s + (−0.775 + 0.391i)15-s + 0.242i·17-s + 0.476i·19-s + (−1.33 + 0.677i)21-s + 0.942·23-s + 0.245·25-s + (0.986 + 0.165i)27-s + 1.37i·29-s − 1.60i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-0.836 - 0.547i$
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: \((2,\ 816,\ (\ :1/2),\ -0.836 - 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163238 + 0.547897i\)
\(L(\frac12)\) \(\approx\) \(0.163238 + 0.547897i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.781 + 1.54i)T \)
17 \( 1 - iT \)
good5 \( 1 + 1.94iT - 5T^{2} \)
7 \( 1 + 3.97iT - 7T^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 + 1.25T + 13T^{2} \)
19 \( 1 - 2.07iT - 19T^{2} \)
23 \( 1 - 4.52T + 23T^{2} \)
29 \( 1 - 7.39iT - 29T^{2} \)
31 \( 1 + 8.94iT - 31T^{2} \)
37 \( 1 - 1.33T + 37T^{2} \)
41 \( 1 + 0.747iT - 41T^{2} \)
43 \( 1 - 4.26iT - 43T^{2} \)
47 \( 1 + 4.99T + 47T^{2} \)
53 \( 1 + 1.80iT - 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 14.1iT - 67T^{2} \)
71 \( 1 + 0.151T + 71T^{2} \)
73 \( 1 + 1.15T + 73T^{2} \)
79 \( 1 - 3.97iT - 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 17.9iT - 89T^{2} \)
97 \( 1 + 9.97T + 97T^{2} \)
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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

−9.920831843202922134694678168950, −8.682655836286810371666173194971, −7.70654351895453779306184861151, −7.44395396485919816261462591057, −6.27672936860880598273796301054, −5.18517564775566755612169604566, −4.57501158360771434342608278359, −3.02059833878716959669040724624, −1.50226198691380108274146900908, −0.29522769987639056554866472447, 2.61947654591466063050910776735, 3.03664202538270229799807820819, 4.74921346721112019500782843606, 5.35643286747925451856447648331, 6.19004996715686340001028411778, 7.23490835795550051455117839047, 8.383329061909244576242653843328, 9.144625253644791131302670783413, 10.03161644523068365861552316829, 10.70684626982784804613108716733

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