Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7 \cdot 601 $
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 + 8-s + 9-s − 10-s − 2·11-s + 12-s + 14-s − 15-s + 16-s + 4·17-s + 18-s + 6·19-s − 20-s + 21-s − 2·22-s + 3·23-s + 24-s − 4·25-s + 27-s + 28-s + 2·29-s − 30-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 + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 25242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 25242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25242\)    =    \(2 \cdot 3 \cdot 7 \cdot 601\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{25242} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 25242,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.196725424$
$L(\frac12)$  $\approx$  $5.196725424$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7,\;601\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;601\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
601 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.40627011346761, −14.59847198644234, −14.39543095996447, −13.79771911184653, −13.25391589935274, −12.79813996276828, −12.10730009542352, −11.65982649581201, −11.24568768843637, −10.37509818390062, −10.04934139734870, −9.287973003190630, −8.712428078195466, −7.827859495620335, −7.614050244456849, −7.249742908418365, −6.077888643095739, −5.790384031496747, −4.903327151680186, −4.470047242745831, −3.748243635203316, −2.959529541128039, −2.699408847650457, −1.568774578473078, −0.8198843654055304, 0.8198843654055304, 1.568774578473078, 2.699408847650457, 2.959529541128039, 3.748243635203316, 4.470047242745831, 4.903327151680186, 5.790384031496747, 6.077888643095739, 7.249742908418365, 7.614050244456849, 7.827859495620335, 8.712428078195466, 9.287973003190630, 10.04934139734870, 10.37509818390062, 11.24568768843637, 11.65982649581201, 12.10730009542352, 12.79813996276828, 13.25391589935274, 13.79771911184653, 14.39543095996447, 14.59847198644234, 15.40627011346761

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