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  + 1.61·2-s + 3-s + 0.618·4-s + 1.61·6-s + 7-s − 2.23·8-s − 2·9-s − 4.47·11-s + 0.618·12-s + 4.23·13-s + 1.61·14-s − 4.85·16-s − 3.23·18-s − 2.76·19-s + 21-s − 7.23·22-s + 23-s − 2.23·24-s + 6.85·26-s − 5·27-s + 0.618·28-s + 7.47·29-s − 9·31-s − 3.38·32-s − 4.47·33-s − 1.23·36-s − 7.70·37-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.660·6-s + 0.377·7-s − 0.790·8-s − 0.666·9-s − 1.34·11-s + 0.178·12-s + 1.17·13-s + 0.432·14-s − 1.21·16-s − 0.762·18-s − 0.634·19-s + 0.218·21-s − 1.54·22-s + 0.208·23-s − 0.456·24-s + 1.34·26-s − 0.962·27-s + 0.116·28-s + 1.38·29-s − 1.61·31-s − 0.597·32-s − 0.778·33-s − 0.206·36-s − 1.26·37-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 \( 1 - 1.61T + 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
13 \( 1 - 4.23T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 2.76T + 19T^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 + 7.70T + 37T^{2} \)
41 \( 1 - 2.23T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + 5.47T + 47T^{2} \)
53 \( 1 + 6.76T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 5.76T + 71T^{2} \)
73 \( 1 + 6.70T + 73T^{2} \)
79 \( 1 + 2.76T + 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 + 8.94T + 89T^{2} \)
97 \( 1 - 9.70T + 97T^{2} \)
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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

−8.125933760229292172296227731292, −7.39590964487500001541968724071, −6.23876918012109201011593600370, −5.77890108494033546614824547625, −4.98868513729720291122205823140, −4.31628866076734236043525363972, −3.30668076776429185708201866173, −2.89752439716216048790040016075, −1.82352050973926868699688820598, 0, 1.82352050973926868699688820598, 2.89752439716216048790040016075, 3.30668076776429185708201866173, 4.31628866076734236043525363972, 4.98868513729720291122205823140, 5.77890108494033546614824547625, 6.23876918012109201011593600370, 7.39590964487500001541968724071, 8.125933760229292172296227731292

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