Properties

Degree 2
Conductor $ 5^{2} \cdot 7 \cdot 23 $
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 − 4-s + 7-s − 3·8-s − 3·9-s + 2·11-s − 4·13-s + 14-s − 16-s + 6·17-s − 3·18-s − 8·19-s + 2·22-s − 23-s − 4·26-s − 28-s + 10·29-s + 10·31-s + 5·32-s + 6·34-s + 3·36-s − 8·37-s − 8·38-s − 2·41-s − 2·44-s − 46-s − 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s + 0.603·11-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.83·19-s + 0.426·22-s − 0.208·23-s − 0.784·26-s − 0.188·28-s + 1.85·29-s + 1.79·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s − 1.31·37-s − 1.29·38-s − 0.312·41-s − 0.301·44-s − 0.147·46-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.793527565$
$L(\frac12)$  $\approx$  $1.793527565$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−17.89902136835414, −17.33010039214084, −17.15020474005605, −16.33852498998087, −15.32805347886893, −14.80598316001591, −14.28122462306671, −14.04072165108404, −13.18303095240544, −12.31889401363248, −12.05241851932183, −11.50622795161756, −10.27818074074263, −10.03209316112717, −8.983335752854459, −8.393376404030028, −7.970958775519788, −6.612458624116954, −6.257527422721416, −5.152608681437976, −4.865865592308691, −3.908898144464231, −3.095142491921366, −2.244196511532632, −0.6753660668505041, 0.6753660668505041, 2.244196511532632, 3.095142491921366, 3.908898144464231, 4.865865592308691, 5.152608681437976, 6.257527422721416, 6.612458624116954, 7.970958775519788, 8.393376404030028, 8.983335752854459, 10.03209316112717, 10.27818074074263, 11.50622795161756, 12.05241851932183, 12.31889401363248, 13.18303095240544, 14.04072165108404, 14.28122462306671, 14.80598316001591, 15.32805347886893, 16.33852498998087, 17.15020474005605, 17.33010039214084, 17.89902136835414

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