Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.56·3-s − 4-s + 1.56·5-s + 1.56·6-s + 3·8-s − 0.561·9-s − 1.56·10-s − 2·11-s + 1.56·12-s − 0.438·13-s − 2.43·15-s − 16-s − 17-s + 0.561·18-s + 7.12·19-s − 1.56·20-s + 2·22-s + 2·23-s − 4.68·24-s − 2.56·25-s + 0.438·26-s + 5.56·27-s + 3.12·29-s + 2.43·30-s − 6·31-s − 5·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.901·3-s − 0.5·4-s + 0.698·5-s + 0.637·6-s + 1.06·8-s − 0.187·9-s − 0.493·10-s − 0.603·11-s + 0.450·12-s − 0.121·13-s − 0.629·15-s − 0.250·16-s − 0.242·17-s + 0.132·18-s + 1.63·19-s − 0.349·20-s + 0.426·22-s + 0.417·23-s − 0.956·24-s − 0.512·25-s + 0.0859·26-s + 1.07·27-s + 0.579·29-s + 0.445·30-s − 1.07·31-s − 0.883·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{731} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 - 1.56T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 0.438T + 13T^{2} \)
19 \( 1 - 7.12T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 5.56T + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
47 \( 1 + 9.56T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 0.684T + 61T^{2} \)
67 \( 1 - 5.56T + 67T^{2} \)
71 \( 1 - 5.56T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + 0.246T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.888980249552727850051448226179, −9.308738138518385212077246178636, −8.325681979355988572320170938278, −7.45534630212084004733258656458, −6.37865224481598929073326882895, −5.31308685634011262394320915221, −4.91376999066276333896195008106, −3.23433596067551510250460117016, −1.54754060973099887411111284050, 0, 1.54754060973099887411111284050, 3.23433596067551510250460117016, 4.91376999066276333896195008106, 5.31308685634011262394320915221, 6.37865224481598929073326882895, 7.45534630212084004733258656458, 8.325681979355988572320170938278, 9.308738138518385212077246178636, 9.888980249552727850051448226179

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