Properties

Label 2-55e2-1.1-c1-0-122
Degree $2$
Conductor $3025$
Sign $-1$
Analytic cond. $24.1547$
Root an. cond. $4.91474$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s − 2·3-s + 0.999·4-s − 3.46·6-s + 3.46·7-s − 1.73·8-s + 9-s − 1.99·12-s + 5.99·14-s − 5·16-s − 6.92·17-s + 1.73·18-s + 6.92·19-s − 6.92·21-s − 6·23-s + 3.46·24-s + 4·27-s + 3.46·28-s + 4·31-s − 5.19·32-s − 11.9·34-s + 0.999·36-s − 10·37-s + 11.9·38-s − 6.92·41-s − 11.9·42-s − 3.46·43-s + ⋯
L(s)  = 1  + 1.22·2-s − 1.15·3-s + 0.499·4-s − 1.41·6-s + 1.30·7-s − 0.612·8-s + 0.333·9-s − 0.577·12-s + 1.60·14-s − 1.25·16-s − 1.68·17-s + 0.408·18-s + 1.58·19-s − 1.51·21-s − 1.25·23-s + 0.707·24-s + 0.769·27-s + 0.654·28-s + 0.718·31-s − 0.918·32-s − 2.05·34-s + 0.166·36-s − 1.64·37-s + 1.94·38-s − 1.08·41-s − 1.85·42-s − 0.528·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(24.1547\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 - 1.73T + 2T^{2} \)
3 \( 1 + 2T + 3T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 - 6.92T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.366233693642661372013880388151, −7.28084559401010097618895313607, −6.54471988288324646127848098086, −5.76380550638857076846614545834, −5.17437584526464953764901082216, −4.67889960416948878982598264478, −3.93988461365387891873046069700, −2.75626991463482279653623135928, −1.58699312365382491621906581844, 0, 1.58699312365382491621906581844, 2.75626991463482279653623135928, 3.93988461365387891873046069700, 4.67889960416948878982598264478, 5.17437584526464953764901082216, 5.76380550638857076846614545834, 6.54471988288324646127848098086, 7.28084559401010097618895313607, 8.366233693642661372013880388151

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