Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 19 $
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 − 3·3-s + 4-s − 5-s − 3·6-s − 5·7-s + 8-s + 6·9-s − 10-s − 4·11-s − 3·12-s − 13-s − 5·14-s + 3·15-s + 16-s − 3·17-s + 6·18-s + 19-s − 20-s + 15·21-s − 4·22-s + 7·23-s − 3·24-s + 25-s − 26-s − 9·27-s − 5·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s − 1.88·7-s + 0.353·8-s + 2·9-s − 0.316·10-s − 1.20·11-s − 0.866·12-s − 0.277·13-s − 1.33·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s + 0.229·19-s − 0.223·20-s + 3.27·21-s − 0.852·22-s + 1.45·23-s − 0.612·24-s + 1/5·25-s − 0.196·26-s − 1.73·27-s − 0.944·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(190\)    =    \(2 \cdot 5 \cdot 19\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 190,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.04947810000832, −18.43242114407029, −17.17261408102034, −16.52997980573399, −15.74211359119195, −15.37995143646139, −13.46605143277111, −12.76226188617008, −12.32630247841887, −11.11517582423131, −10.53685995532110, −9.431192927280395, −7.343141243954329, −6.590129766673766, −5.667681214154848, −4.671019258233202, −3.150155860520488, 0, 3.150155860520488, 4.671019258233202, 5.667681214154848, 6.590129766673766, 7.343141243954329, 9.431192927280395, 10.53685995532110, 11.11517582423131, 12.32630247841887, 12.76226188617008, 13.46605143277111, 15.37995143646139, 15.74211359119195, 16.52997980573399, 17.17261408102034, 18.43242114407029, 19.04947810000832

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