Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 19^{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-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 12-s + 6·13-s − 14-s − 15-s − 16-s + 2·17-s − 18-s − 20-s − 21-s + 8·23-s − 3·24-s + 25-s − 6·26-s − 27-s − 28-s + 2·29-s + 30-s − 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.182·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 37905 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 37905 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(37905\)    =    \(3 \cdot 5 \cdot 7 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{37905} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 37905,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.744908250$
$L(\frac12)$  $\approx$  $1.744908250$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.78705233716252, −14.23242758877223, −13.82679818238434, −13.20872466353931, −12.84318453096380, −12.33656378619000, −11.41412378323759, −10.97201992185302, −10.74526676282943, −10.11373691364384, −9.429833445183482, −8.971219833515737, −8.653875472723849, −7.831702875521610, −7.465958104264541, −6.700098899212344, −6.047865240230688, −5.555214787473983, −4.930916066063638, −4.347314730166904, −3.679432959398745, −2.912477876318848, −1.810383533305984, −1.162046485282680, −0.6987666088926710, 0.6987666088926710, 1.162046485282680, 1.810383533305984, 2.912477876318848, 3.679432959398745, 4.347314730166904, 4.930916066063638, 5.555214787473983, 6.047865240230688, 6.700098899212344, 7.465958104264541, 7.831702875521610, 8.653875472723849, 8.971219833515737, 9.429833445183482, 10.11373691364384, 10.74526676282943, 10.97201992185302, 11.41412378323759, 12.33656378619000, 12.84318453096380, 13.20872466353931, 13.82679818238434, 14.23242758877223, 14.78705233716252

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