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  + 1.36·2-s + 3.15·3-s − 6.14·4-s + 3.03·5-s + 4.29·6-s − 7.94·7-s − 19.2·8-s − 17.0·9-s + 4.12·10-s + 27.6·11-s − 19.3·12-s + 58.1·13-s − 10.8·14-s + 9.56·15-s + 22.9·16-s − 17·17-s − 23.2·18-s + 89.1·19-s − 18.6·20-s − 25.0·21-s + 37.5·22-s − 115.·23-s − 60.7·24-s − 115.·25-s + 79.1·26-s − 138.·27-s + 48.8·28-s + ⋯
L(s)  = 1  + 0.481·2-s + 0.607·3-s − 0.768·4-s + 0.271·5-s + 0.292·6-s − 0.428·7-s − 0.851·8-s − 0.631·9-s + 0.130·10-s + 0.756·11-s − 0.466·12-s + 1.23·13-s − 0.206·14-s + 0.164·15-s + 0.358·16-s − 0.242·17-s − 0.303·18-s + 1.07·19-s − 0.208·20-s − 0.260·21-s + 0.364·22-s − 1.04·23-s − 0.516·24-s − 0.926·25-s + 0.596·26-s − 0.990·27-s + 0.329·28-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  =  $(2,\ 17,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $1.25973$
$L(\frac12)$  $\approx$  $1.25973$
$L(\frac{5}{2})$   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 \( 1 - 1.36T + 8T^{2} \)
3 \( 1 - 3.15T + 27T^{2} \)
5 \( 1 - 3.03T + 125T^{2} \)
7 \( 1 + 7.94T + 343T^{2} \)
11 \( 1 - 27.6T + 1.33e3T^{2} \)
13 \( 1 - 58.1T + 2.19e3T^{2} \)
19 \( 1 - 89.1T + 6.85e3T^{2} \)
23 \( 1 + 115.T + 1.21e4T^{2} \)
29 \( 1 + 128.T + 2.43e4T^{2} \)
31 \( 1 - 273.T + 2.97e4T^{2} \)
37 \( 1 + 132.T + 5.06e4T^{2} \)
41 \( 1 + 470.T + 6.89e4T^{2} \)
43 \( 1 - 352.T + 7.95e4T^{2} \)
47 \( 1 - 152.T + 1.03e5T^{2} \)
53 \( 1 - 527.T + 1.48e5T^{2} \)
59 \( 1 + 292.T + 2.05e5T^{2} \)
61 \( 1 + 53.8T + 2.26e5T^{2} \)
67 \( 1 - 52.9T + 3.00e5T^{2} \)
71 \( 1 - 788.T + 3.57e5T^{2} \)
73 \( 1 - 295.T + 3.89e5T^{2} \)
79 \( 1 + 720.T + 4.93e5T^{2} \)
83 \( 1 + 116.T + 5.71e5T^{2} \)
89 \( 1 + 813.T + 7.04e5T^{2} \)
97 \( 1 - 794.T + 9.12e5T^{2} \)
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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

−18.47951223268626911027103487299, −17.26837424769575811362802477693, −15.56194346425825622226188432957, −14.05477192401478735713806224842, −13.53903149732419800522825997454, −11.82587820016891111095247821777, −9.632473151912304259544737115312, −8.468797931306279695202548895799, −5.93382345550704463877322310005, −3.63824331318068623675418956800, 3.63824331318068623675418956800, 5.93382345550704463877322310005, 8.468797931306279695202548895799, 9.632473151912304259544737115312, 11.82587820016891111095247821777, 13.53903149732419800522825997454, 14.05477192401478735713806224842, 15.56194346425825622226188432957, 17.26837424769575811362802477693, 18.47951223268626911027103487299

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