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Properties

Label 2-816-17.8-c1-0-3
Degree $2$
Conductor $816$
Sign $0.427 - 0.904i$
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  + (0.382 − 0.923i)3-s + (−2.50 − 1.03i)5-s + (−2.22 + 0.919i)7-s + (−0.707 − 0.707i)9-s + (1.87 + 4.52i)11-s + 1.07i·13-s + (−1.91 + 1.91i)15-s + (3.25 − 2.52i)17-s + (−1.61 + 1.61i)19-s + 2.40i·21-s + (2.41 + 5.82i)23-s + (1.67 + 1.67i)25-s + (−0.923 + 0.382i)27-s + (3.13 + 1.29i)29-s + (−2.67 + 6.46i)31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (−1.12 − 0.464i)5-s + (−0.839 + 0.347i)7-s + (−0.235 − 0.235i)9-s + (0.564 + 1.36i)11-s + 0.296i·13-s + (−0.495 + 0.495i)15-s + (0.789 − 0.613i)17-s + (−0.369 + 0.369i)19-s + 0.524i·21-s + (0.503 + 1.21i)23-s + (0.335 + 0.335i)25-s + (−0.177 + 0.0736i)27-s + (0.582 + 0.241i)29-s + (−0.480 + 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.427 - 0.904i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761664 + 0.482452i\)
\(L(\frac12)\) \(\approx\) \(0.761664 + 0.482452i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (-3.25 + 2.52i)T \)
good5 \( 1 + (2.50 + 1.03i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.22 - 0.919i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.87 - 4.52i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.07iT - 13T^{2} \)
19 \( 1 + (1.61 - 1.61i)T - 19iT^{2} \)
23 \( 1 + (-2.41 - 5.82i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.13 - 1.29i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.67 - 6.46i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.63 - 3.93i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-7.08 + 2.93i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-4.37 - 4.37i)T + 43iT^{2} \)
47 \( 1 + 8.19iT - 47T^{2} \)
53 \( 1 + (7.97 - 7.97i)T - 53iT^{2} \)
59 \( 1 + (-8.89 - 8.89i)T + 59iT^{2} \)
61 \( 1 + (2.52 - 1.04i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + (-1.62 + 3.92i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (9.47 + 3.92i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.464 + 1.12i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (2.92 - 2.92i)T - 83iT^{2} \)
89 \( 1 + 2.96iT - 89T^{2} \)
97 \( 1 + (-16.8 - 6.98i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.26851983135703912030933932162, −9.351867063031905740266913925446, −8.787354961394753468522830127124, −7.59486449615334167073924099105, −7.22070157586181288497206548192, −6.17685768399976610522408300388, −4.93727956028232619167583642927, −3.94189266938669502805769611699, −2.94835628803804232576151998092, −1.38934219449494891185104952713, 0.46907986173361875133489661982, 2.88963548744372233328289978952, 3.60737327096553328572215714633, 4.31508848105023015399172949490, 5.80691515532617449395379977185, 6.58732461479829780804112402748, 7.65021861621831467658767013106, 8.379910543577765416294128662630, 9.210836482600147230622285672982, 10.18820885369788438094812693234

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