Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s + 5-s − 6-s + 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 13-s + 15-s − 16-s + 2·17-s − 18-s − 4·19-s − 20-s − 4·22-s + 8·23-s + 3·24-s + 25-s − 26-s + 27-s − 2·29-s − 30-s − 8·31-s − 5·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.277·13-s + 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 0.612·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.371·29-s − 0.182·30-s − 1.43·31-s − 0.883·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(195\)    =    \(3 \cdot 5 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{195} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 195,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.005712135$
$L(\frac12)$  $\approx$  $1.005712135$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−19.42094759164126, −18.72153782482744, −17.97855785465572, −16.91344958323094, −16.62608053688346, −14.97410239917623, −14.45862846761434, −13.40997026029394, −12.77225124673051, −11.29973861776917, −10.27202750242873, −9.244674050330196, −8.846234947699956, −7.660937674278081, −6.486549668343486, −4.888235951648040, −3.567991924800003, −1.538279464180725, 1.538279464180725, 3.567991924800003, 4.888235951648040, 6.486549668343486, 7.660937674278081, 8.846234947699956, 9.244674050330196, 10.27202750242873, 11.29973861776917, 12.77225124673051, 13.40997026029394, 14.45862846761434, 14.97410239917623, 16.62608053688346, 16.91344958323094, 17.97855785465572, 18.72153782482744, 19.42094759164126

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