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  + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 3·7-s + 9-s + 2·10-s − 5·11-s + 2·12-s + 13-s − 6·14-s + 15-s − 4·16-s + 5·17-s + 2·18-s + 2·19-s + 2·20-s − 3·21-s − 10·22-s − 23-s + 25-s + 2·26-s + 27-s − 6·28-s + 10·29-s + 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 1.13·7-s + 1/3·9-s + 0.632·10-s − 1.50·11-s + 0.577·12-s + 0.277·13-s − 1.60·14-s + 0.258·15-s − 16-s + 1.21·17-s + 0.471·18-s + 0.458·19-s + 0.447·20-s − 0.654·21-s − 2.13·22-s − 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.13·28-s + 1.85·29-s + 0.365·30-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  =  $(2,\ 195,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.479355493$
$L(\frac12)$  $\approx$  $2.479355493$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−19.84622376002264, −18.71865141200414, −18.12012703461090, −16.58779500271098, −15.76338785973071, −15.19693992489799, −13.90812751317431, −13.65371651282918, −12.70705052788882, −12.12132680691426, −10.48774420896021, −9.740914180850663, −8.434475207546954, −7.077558933522130, −5.939543040286531, −5.028031773613241, −3.482004434187194, −2.700481529543811, 2.700481529543811, 3.482004434187194, 5.028031773613241, 5.939543040286531, 7.077558933522130, 8.434475207546954, 9.740914180850663, 10.48774420896021, 12.12132680691426, 12.70705052788882, 13.65371651282918, 13.90812751317431, 15.19693992489799, 15.76338785973071, 16.58779500271098, 18.12012703461090, 18.71865141200414, 19.84622376002264

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