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Properties

Label 2-816-51.38-c0-0-0
Degree $2$
Conductor $816$
Sign $0.788 - 0.615i$
Analytic cond. $0.407237$
Root an. cond. $0.638151$
Motivic weight $0$
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 + (0.707 + 0.707i)5-s + (1 + i)7-s + 1.00i·9-s + (−0.707 + 0.707i)11-s − 13-s − 1.00i·15-s + (−0.707 + 0.707i)17-s + i·19-s − 1.41i·21-s + (0.707 − 0.707i)23-s + (0.707 − 0.707i)27-s + 1.00·33-s + 1.41i·35-s + (1 − i)37-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (1 + i)7-s + 1.00i·9-s + (−0.707 + 0.707i)11-s − 13-s − 1.00i·15-s + (−0.707 + 0.707i)17-s + i·19-s − 1.41i·21-s + (0.707 − 0.707i)23-s + (0.707 − 0.707i)27-s + 1.00·33-s + 1.41i·35-s + (1 − i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(0.407237\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :0),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8776218860\)
\(L(\frac12)\) \(\approx\) \(0.8776218860\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good5 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
7 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (1 + i)T + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.63740004381422615501415811904, −9.926363135279425156389304727944, −8.722211292134738269101345978439, −7.82767533845514218559520864467, −7.07154118030382140335380177828, −6.08665811953861083718863794134, −5.40919383684444560551650610540, −4.55193066319256154341603083746, −2.40551878557157946802460722676, −2.04109928179122818228148675544, 1.04707866778644570645640730399, 2.83560256827709327761061770601, 4.47402707240472082279345130148, 4.88376729235795507800230886911, 5.66262821701748551556640648170, 6.88138249481557852209105039249, 7.76711876309596233183770385002, 8.903853738184397068743815816261, 9.590871980742969288491011660424, 10.36527564722934166380455439910

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