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Properties

Label 2-816-16.13-c1-0-1
Degree $2$
Conductor $816$
Sign $-0.981 - 0.190i$
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  + (−0.606 − 1.27i)2-s + (0.707 + 0.707i)3-s + (−1.26 + 1.54i)4-s + (−2.11 + 2.11i)5-s + (0.474 − 1.33i)6-s + 1.09i·7-s + (2.74 + 0.678i)8-s + 1.00i·9-s + (3.97 + 1.41i)10-s + (−1.99 + 1.99i)11-s + (−1.98 + 0.200i)12-s + (−1.10 − 1.10i)13-s + (1.40 − 0.664i)14-s − 2.98·15-s + (−0.797 − 3.91i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.428 − 0.903i)2-s + (0.408 + 0.408i)3-s + (−0.632 + 0.774i)4-s + (−0.943 + 0.943i)5-s + (0.193 − 0.543i)6-s + 0.414i·7-s + (0.970 + 0.239i)8-s + 0.333i·9-s + (1.25 + 0.448i)10-s + (−0.600 + 0.600i)11-s + (−0.574 + 0.0578i)12-s + (−0.306 − 0.306i)13-s + (0.374 − 0.177i)14-s − 0.770·15-s + (−0.199 − 0.979i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-0.981 - 0.190i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ -0.981 - 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0151253 + 0.157108i\)
\(L(\frac12)\) \(\approx\) \(0.0151253 + 0.157108i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.606 + 1.27i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + T \)
good5 \( 1 + (2.11 - 2.11i)T - 5iT^{2} \)
7 \( 1 - 1.09iT - 7T^{2} \)
11 \( 1 + (1.99 - 1.99i)T - 11iT^{2} \)
13 \( 1 + (1.10 + 1.10i)T + 13iT^{2} \)
19 \( 1 + (2.88 + 2.88i)T + 19iT^{2} \)
23 \( 1 + 5.91iT - 23T^{2} \)
29 \( 1 + (5.97 + 5.97i)T + 29iT^{2} \)
31 \( 1 - 4.00T + 31T^{2} \)
37 \( 1 + (-4.30 + 4.30i)T - 37iT^{2} \)
41 \( 1 - 6.79iT - 41T^{2} \)
43 \( 1 + (1.16 - 1.16i)T - 43iT^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (7.88 - 7.88i)T - 53iT^{2} \)
59 \( 1 + (4.50 - 4.50i)T - 59iT^{2} \)
61 \( 1 + (9.58 + 9.58i)T + 61iT^{2} \)
67 \( 1 + (-0.299 - 0.299i)T + 67iT^{2} \)
71 \( 1 + 2.77iT - 71T^{2} \)
73 \( 1 - 5.26iT - 73T^{2} \)
79 \( 1 - 4.62T + 79T^{2} \)
83 \( 1 + (-9.65 - 9.65i)T + 83iT^{2} \)
89 \( 1 + 5.02iT - 89T^{2} \)
97 \( 1 + 1.23T + 97T^{2} \)
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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

−10.77032986980479342764743016645, −9.867170529727354535368355765479, −9.132552318510537234150966942656, −8.034088714257574904310685293775, −7.68687013980333515386945909723, −6.49972467917185486638676225648, −4.82830569772332236329789232371, −4.10548294688242098107444917516, −2.94801123534936860748281688573, −2.34285232173482734415240968081, 0.085866673215736627628071561767, 1.53327313423752544845289154303, 3.54974120985504516237796224267, 4.53479292370338636464492997227, 5.45488777876560764824936364201, 6.56730628924688749968782349220, 7.56734138480027114347456453604, 8.018281589673095652676926117085, 8.743451420891182775483172184555, 9.489381761735056172599836251526

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