Properties

Degree 2
Conductor $ 3 \cdot 5^{2} $
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 − 2·6-s − 3·7-s + 9-s + 2·11-s − 2·12-s + 13-s − 6·14-s − 4·16-s + 2·17-s + 2·18-s − 5·19-s + 3·21-s + 4·22-s + 6·23-s + 2·26-s − 27-s − 6·28-s + 10·29-s − 3·31-s − 8·32-s − 2·33-s + 4·34-s + 2·36-s + 2·37-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 1.13·7-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 0.277·13-s − 1.60·14-s − 16-s + 0.485·17-s + 0.471·18-s − 1.14·19-s + 0.654·21-s + 0.852·22-s + 1.25·23-s + 0.392·26-s − 0.192·27-s − 1.13·28-s + 1.85·29-s − 0.538·31-s − 1.41·32-s − 0.348·33-s + 0.685·34-s + 1/3·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(75\)    =    \(3 \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{75} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 75,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.40253$
$L(\frac12)$  $\approx$  $1.40253$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
5 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−14.43096345053950877210365893366, −13.27531306639888978122196489064, −12.60092562784261018149461459363, −11.69351541999948731726714975337, −10.39882094959826449711333942129, −8.979440455637625961269202062502, −6.79375443883521821659151920822, −6.05569215012514529764032801313, −4.62309319523854591570832526017, −3.24466149791278329969031792942, 3.24466149791278329969031792942, 4.62309319523854591570832526017, 6.05569215012514529764032801313, 6.79375443883521821659151920822, 8.979440455637625961269202062502, 10.39882094959826449711333942129, 11.69351541999948731726714975337, 12.60092562784261018149461459363, 13.27531306639888978122196489064, 14.43096345053950877210365893366

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