Properties

Degree 2
Conductor $ 2 \cdot 15739 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 3·5-s + 2·6-s − 4·7-s − 8-s + 9-s + 3·10-s − 2·11-s − 2·12-s − 13-s + 4·14-s + 6·15-s + 16-s − 6·17-s − 18-s − 4·19-s − 3·20-s + 8·21-s + 2·22-s − 9·23-s + 2·24-s + 4·25-s + 26-s + 4·27-s − 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.603·11-s − 0.577·12-s − 0.277·13-s + 1.06·14-s + 1.54·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s + 1.74·21-s + 0.426·22-s − 1.87·23-s + 0.408·24-s + 4/5·25-s + 0.196·26-s + 0.769·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31478 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31478 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31478\)    =    \(2 \cdot 15739\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{31478} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 31478,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;15739\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;15739\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
15739 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.96157834516836, −15.69435472291278, −15.08103436267391, −14.52630891697278, −13.55301720842949, −12.85627930498837, −12.67684776592718, −12.03522575440884, −11.49584059895362, −11.15244581201941, −10.54823120832840, −10.12938795028159, −9.496885177769871, −8.765728940046544, −8.363966065104031, −7.604326513668194, −7.069084585578214, −6.557613375643603, −6.125970678169867, −5.408232948361362, −4.719651867365276, −3.757255540084210, −3.598924365508598, −2.504043641997019, −1.773887930464119, 0, 0, 0, 1.773887930464119, 2.504043641997019, 3.598924365508598, 3.757255540084210, 4.719651867365276, 5.408232948361362, 6.125970678169867, 6.557613375643603, 7.069084585578214, 7.604326513668194, 8.363966065104031, 8.765728940046544, 9.496885177769871, 10.12938795028159, 10.54823120832840, 11.15244581201941, 11.49584059895362, 12.03522575440884, 12.67684776592718, 12.85627930498837, 13.55301720842949, 14.52630891697278, 15.08103436267391, 15.69435472291278, 15.96157834516836

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