Properties

Degree 2
Conductor $ 5^{2} \cdot 7 \cdot 23 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s + 3-s − 1.61·4-s − 0.618·6-s + 7-s + 2.23·8-s − 2·9-s + 4.47·11-s − 1.61·12-s − 0.236·13-s − 0.618·14-s + 1.85·16-s + 1.23·18-s − 7.23·19-s + 21-s − 2.76·22-s + 23-s + 2.23·24-s + 0.145·26-s − 5·27-s − 1.61·28-s − 1.47·29-s − 9·31-s − 5.61·32-s + 4.47·33-s + 3.23·36-s + 5.70·37-s + ⋯
L(s)  = 1  − 0.437·2-s + 0.577·3-s − 0.809·4-s − 0.252·6-s + 0.377·7-s + 0.790·8-s − 0.666·9-s + 1.34·11-s − 0.467·12-s − 0.0654·13-s − 0.165·14-s + 0.463·16-s + 0.291·18-s − 1.66·19-s + 0.218·21-s − 0.589·22-s + 0.208·23-s + 0.456·24-s + 0.0286·26-s − 0.962·27-s − 0.305·28-s − 0.273·29-s − 1.61·31-s − 0.993·32-s + 0.778·33-s + 0.539·36-s + 0.938·37-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  :  $\chi_{4025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 4025,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.618T + 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 0.236T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
29 \( 1 + 1.47T + 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 - 5.70T + 37T^{2} \)
41 \( 1 + 2.23T + 41T^{2} \)
43 \( 1 + 2.47T + 43T^{2} \)
47 \( 1 - 3.47T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 1.52T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 + 7.23T + 79T^{2} \)
83 \( 1 + 6.47T + 83T^{2} \)
89 \( 1 - 8.94T + 89T^{2} \)
97 \( 1 + 3.70T + 97T^{2} \)
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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

−8.350329933734952409859327177506, −7.59722754166431819530618122077, −6.70549368654592222642490328159, −5.86541483197722003307724598324, −4.98968771633073778616943756383, −4.08238891614911344155956974363, −3.61938829907485201539472834940, −2.32208641849278377881195589094, −1.39833687032560782469157297142, 0, 1.39833687032560782469157297142, 2.32208641849278377881195589094, 3.61938829907485201539472834940, 4.08238891614911344155956974363, 4.98968771633073778616943756383, 5.86541483197722003307724598324, 6.70549368654592222642490328159, 7.59722754166431819530618122077, 8.350329933734952409859327177506

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