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  − 5.03·2-s + 8.47·3-s + 17.3·4-s + 0.885·5-s − 42.6·6-s + 3.81·7-s − 46.9·8-s + 44.8·9-s − 4.45·10-s − 52.3·11-s + 146.·12-s − 8.06·13-s − 19.2·14-s + 7.50·15-s + 97.5·16-s − 17·17-s − 225.·18-s − 66.5·19-s + 15.3·20-s + 32.3·21-s + 263.·22-s + 180.·23-s − 397.·24-s − 124.·25-s + 40.5·26-s + 151.·27-s + 66.1·28-s + ⋯
L(s)  = 1  − 1.77·2-s + 1.63·3-s + 2.16·4-s + 0.0792·5-s − 2.90·6-s + 0.206·7-s − 2.07·8-s + 1.66·9-s − 0.140·10-s − 1.43·11-s + 3.53·12-s − 0.171·13-s − 0.366·14-s + 0.129·15-s + 1.52·16-s − 0.242·17-s − 2.95·18-s − 0.803·19-s + 0.171·20-s + 0.336·21-s + 2.55·22-s + 1.63·23-s − 3.38·24-s − 0.993·25-s + 0.305·26-s + 1.07·27-s + 0.446·28-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  =  $(2,\ 17,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $0.757110$
$L(\frac12)$  $\approx$  $0.757110$
$L(\frac{5}{2})$   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 \( 1 + 5.03T + 8T^{2} \)
3 \( 1 - 8.47T + 27T^{2} \)
5 \( 1 - 0.885T + 125T^{2} \)
7 \( 1 - 3.81T + 343T^{2} \)
11 \( 1 + 52.3T + 1.33e3T^{2} \)
13 \( 1 + 8.06T + 2.19e3T^{2} \)
19 \( 1 + 66.5T + 6.85e3T^{2} \)
23 \( 1 - 180.T + 1.21e4T^{2} \)
29 \( 1 + 41.2T + 2.43e4T^{2} \)
31 \( 1 + 34.9T + 2.97e4T^{2} \)
37 \( 1 - 130.T + 5.06e4T^{2} \)
41 \( 1 + 17.9T + 6.89e4T^{2} \)
43 \( 1 - 277.T + 7.95e4T^{2} \)
47 \( 1 - 463.T + 1.03e5T^{2} \)
53 \( 1 + 329.T + 1.48e5T^{2} \)
59 \( 1 - 678.T + 2.05e5T^{2} \)
61 \( 1 - 340.T + 2.26e5T^{2} \)
67 \( 1 - 15.3T + 3.00e5T^{2} \)
71 \( 1 + 670.T + 3.57e5T^{2} \)
73 \( 1 - 193.T + 3.89e5T^{2} \)
79 \( 1 - 1.08e3T + 4.93e5T^{2} \)
83 \( 1 + 865.T + 5.71e5T^{2} \)
89 \( 1 - 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 379.T + 9.12e5T^{2} \)
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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

−18.73634788817423347250629986430, −17.48225705643740613688541171471, −15.88180455081542440268242258296, −14.91646528533242250020355845294, −13.15157710861435480148132838771, −10.78651734741238135136874699464, −9.496516817172455924137662028802, −8.397324407657249551110908524756, −7.42880277196023059009466302805, −2.43281623707147080051052305557, 2.43281623707147080051052305557, 7.42880277196023059009466302805, 8.397324407657249551110908524756, 9.496516817172455924137662028802, 10.78651734741238135136874699464, 13.15157710861435480148132838771, 14.91646528533242250020355845294, 15.88180455081542440268242258296, 17.48225705643740613688541171471, 18.73634788817423347250629986430

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