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Properties

Label 2-17-17.16-c9-0-2
Degree $2$
Conductor $17$
Sign $-0.0396 - 0.999i$
Analytic cond. $8.75560$
Root an. cond. $2.95898$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 36.0·2-s + 119. i·3-s + 785.·4-s − 917. i·5-s − 4.31e3i·6-s − 4.19e3i·7-s − 9.83e3·8-s + 5.29e3·9-s + 3.30e4i·10-s + 5.60e4i·11-s + 9.41e4i·12-s + 1.94e3·13-s + 1.51e5i·14-s + 1.10e5·15-s − 4.77e4·16-s + (1.36e4 + 3.44e5i)17-s + ⋯
L(s)  = 1  − 1.59·2-s + 0.854i·3-s + 1.53·4-s − 0.656i·5-s − 1.36i·6-s − 0.660i·7-s − 0.849·8-s + 0.269·9-s + 1.04i·10-s + 1.15i·11-s + 1.31i·12-s + 0.0188·13-s + 1.05i·14-s + 0.561·15-s − 0.181·16-s + (0.0396 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0396 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0396 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.0396 - 0.999i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.0396 - 0.999i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.455092 + 0.473489i\)
\(L(\frac12)\) \(\approx\) \(0.455092 + 0.473489i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.36e4 - 3.44e5i)T \)
good2 \( 1 + 36.0T + 512T^{2} \)
3 \( 1 - 119. iT - 1.96e4T^{2} \)
5 \( 1 + 917. iT - 1.95e6T^{2} \)
7 \( 1 + 4.19e3iT - 4.03e7T^{2} \)
11 \( 1 - 5.60e4iT - 2.35e9T^{2} \)
13 \( 1 - 1.94e3T + 1.06e10T^{2} \)
19 \( 1 + 3.48e5T + 3.22e11T^{2} \)
23 \( 1 - 2.87e5iT - 1.80e12T^{2} \)
29 \( 1 - 3.74e6iT - 1.45e13T^{2} \)
31 \( 1 - 3.22e6iT - 2.64e13T^{2} \)
37 \( 1 + 7.92e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.96e7iT - 3.27e14T^{2} \)
43 \( 1 - 1.54e7T + 5.02e14T^{2} \)
47 \( 1 + 2.94e7T + 1.11e15T^{2} \)
53 \( 1 - 1.12e8T + 3.29e15T^{2} \)
59 \( 1 + 1.42e8T + 8.66e15T^{2} \)
61 \( 1 - 2.77e7iT - 1.16e16T^{2} \)
67 \( 1 + 8.42e7T + 2.72e16T^{2} \)
71 \( 1 - 3.54e8iT - 4.58e16T^{2} \)
73 \( 1 - 2.56e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.39e8iT - 1.19e17T^{2} \)
83 \( 1 - 5.29e8T + 1.86e17T^{2} \)
89 \( 1 + 6.33e8T + 3.50e17T^{2} \)
97 \( 1 + 1.00e9iT - 7.60e17T^{2} \)
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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

−17.02442895781229302313704903546, −16.26050028726272398561975173203, −15.00028817918793086183958062724, −12.73735519942449449748869917216, −10.73977799147965137468144529917, −9.906885629583010355067894633430, −8.725178007625868243995133898232, −7.17943107204821916182933941587, −4.44070299484765887180454511503, −1.39153181930350342557709442193, 0.60389583737269171306815274619, 2.37608635176609514704847855056, 6.42809019091976398040049055926, 7.69620147469517769821876965498, 8.998853366636255413905343810494, 10.56226295141135683643670229909, 11.84326812816295253468522694512, 13.64192780567591930123926618320, 15.48477185383296527872471906195, 16.77749361537817156053449949703

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