Properties

Degree 2
Conductor 17
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.67·2-s − 7.62·3-s + 13.8·4-s − 11.9·5-s − 35.6·6-s + 26.1·7-s + 27.1·8-s + 31.2·9-s − 55.6·10-s − 3.24·11-s − 105.·12-s − 20.0·13-s + 122.·14-s + 90.9·15-s + 16.4·16-s − 17·17-s + 145.·18-s + 57.3·19-s − 164.·20-s − 199.·21-s − 15.1·22-s + 77.0·23-s − 207.·24-s + 17.0·25-s − 93.6·26-s − 32.1·27-s + 361.·28-s + ⋯
L(s)  = 1  + 1.65·2-s − 1.46·3-s + 1.72·4-s − 1.06·5-s − 2.42·6-s + 1.41·7-s + 1.20·8-s + 1.15·9-s − 1.76·10-s − 0.0889·11-s − 2.53·12-s − 0.427·13-s + 2.32·14-s + 1.56·15-s + 0.257·16-s − 0.242·17-s + 1.90·18-s + 0.692·19-s − 1.84·20-s − 2.07·21-s − 0.146·22-s + 0.698·23-s − 1.76·24-s + 0.136·25-s − 0.706·26-s − 0.229·27-s + 2.43·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{17} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 17,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $1.47255$
$L(\frac12)$  $\approx$  $1.47255$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 17$, \(F_p\) is a polynomial of degree 2. If $p = 17$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad17 \( 1 + 17T \)
good2 \( 1 - 4.67T + 8T^{2} \)
3 \( 1 + 7.62T + 27T^{2} \)
5 \( 1 + 11.9T + 125T^{2} \)
7 \( 1 - 26.1T + 343T^{2} \)
11 \( 1 + 3.24T + 1.33e3T^{2} \)
13 \( 1 + 20.0T + 2.19e3T^{2} \)
19 \( 1 - 57.3T + 6.85e3T^{2} \)
23 \( 1 - 77.0T + 1.21e4T^{2} \)
29 \( 1 + 286.T + 2.43e4T^{2} \)
31 \( 1 + 8.54T + 2.97e4T^{2} \)
37 \( 1 - 357.T + 5.06e4T^{2} \)
41 \( 1 - 194.T + 6.89e4T^{2} \)
43 \( 1 + 74.2T + 7.95e4T^{2} \)
47 \( 1 - 23.6T + 1.03e5T^{2} \)
53 \( 1 - 104.T + 1.48e5T^{2} \)
59 \( 1 - 249.T + 2.05e5T^{2} \)
61 \( 1 + 370.T + 2.26e5T^{2} \)
67 \( 1 - 939.T + 3.00e5T^{2} \)
71 \( 1 + 520.T + 3.57e5T^{2} \)
73 \( 1 - 348.T + 3.89e5T^{2} \)
79 \( 1 + 953.T + 4.93e5T^{2} \)
83 \( 1 + 1.41e3T + 5.71e5T^{2} \)
89 \( 1 + 486.T + 7.04e5T^{2} \)
97 \( 1 + 685.T + 9.12e5T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.23764392374568123311530356387, −16.84478912715586378870444349204, −15.56017278704067734646266647191, −14.54293188176067705734753133801, −12.77482808723179505181256238678, −11.47528142904054404240864570025, −11.30398669605301579176000431340, −7.40099051835665882197576801088, −5.51674445038489297870992546733, −4.40891902220805417967654927496, 4.40891902220805417967654927496, 5.51674445038489297870992546733, 7.40099051835665882197576801088, 11.30398669605301579176000431340, 11.47528142904054404240864570025, 12.77482808723179505181256238678, 14.54293188176067705734753133801, 15.56017278704067734646266647191, 16.84478912715586378870444349204, 18.23764392374568123311530356387

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