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Properties

Label 2-816-17.16-c1-0-3
Degree $2$
Conductor $816$
Sign $0.242 - 0.970i$
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  − i·3-s + 3i·5-s − 2i·7-s − 9-s + 5i·11-s − 13-s + 3·15-s + (4 + i)17-s − 5·19-s − 2·21-s + i·23-s − 4·25-s + i·27-s + 6i·29-s + 10i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.34i·5-s − 0.755i·7-s − 0.333·9-s + 1.50i·11-s − 0.277·13-s + 0.774·15-s + (0.970 + 0.242i)17-s − 1.14·19-s − 0.436·21-s + 0.208i·23-s − 0.800·25-s + 0.192i·27-s + 1.11i·29-s + 1.79i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990857 + 0.773638i\)
\(L(\frac12)\) \(\approx\) \(0.990857 + 0.773638i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
17 \( 1 + (-4 - i)T \)
good5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.45832679511651314997877971663, −9.864590189859577582501907147927, −8.610510041652422717039870300437, −7.43972904483132463953351437923, −7.11349142617662292711308784887, −6.40914557841078495951693162589, −5.10223503607140180997399836594, −3.89730163877048501169750124055, −2.82891727314457078693606793740, −1.65394988211164992103358705270, 0.62880148771316615454071715260, 2.41238201414587022170963397240, 3.73391526919344662324467991881, 4.71259043592036137598669427336, 5.61763308646537471282136554031, 6.16659899187124621080955562829, 7.951794712708468396030150483731, 8.415860271391059508335013084470, 9.210946037586103675852244490224, 9.818703722833984388435045176296

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