Properties

Label 2-55e2-1.1-c1-0-135
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  − 0.517·2-s + 1.93·3-s − 1.73·4-s − 0.999·6-s + 3.34·7-s + 1.93·8-s + 0.732·9-s − 3.34·12-s − 4.24·13-s − 1.73·14-s + 2.46·16-s − 3.86·17-s − 0.378·18-s − 4.19·19-s + 6.46·21-s − 3.48·23-s + 3.73·24-s + 2.19·26-s − 4.38·27-s − 5.79·28-s + 6.92·29-s − 8.73·31-s − 5.13·32-s + 1.99·34-s − 1.26·36-s − 1.79·37-s + 2.17·38-s + ⋯
L(s)  = 1  − 0.366·2-s + 1.11·3-s − 0.866·4-s − 0.408·6-s + 1.26·7-s + 0.683·8-s + 0.244·9-s − 0.965·12-s − 1.17·13-s − 0.462·14-s + 0.616·16-s − 0.937·17-s − 0.0893·18-s − 0.962·19-s + 1.41·21-s − 0.726·23-s + 0.761·24-s + 0.430·26-s − 0.843·27-s − 1.09·28-s + 1.28·29-s − 1.56·31-s − 0.908·32-s + 0.342·34-s − 0.211·36-s − 0.294·37-s + 0.352·38-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 \( 1 + 0.517T + 2T^{2} \)
3 \( 1 - 1.93T + 3T^{2} \)
7 \( 1 - 3.34T + 7T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 3.86T + 17T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 + 3.48T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + 1.79T + 37T^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 - 6.45T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 2.17T + 53T^{2} \)
59 \( 1 + 1.26T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 2.20T + 67T^{2} \)
71 \( 1 + 8.19T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 - 1.46T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 0.464T + 89T^{2} \)
97 \( 1 - 9.14T + 97T^{2} \)
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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

−8.395439613379677401257080046208, −7.87161063317590005101558937033, −7.28817140568942164865320592150, −6.04718050972638017506347344742, −4.87065562354408528436738238454, −4.55164923330421899238332547875, −3.58220034193358197913409908408, −2.38845495692263852713418227376, −1.70421975595231102591737677079, 0, 1.70421975595231102591737677079, 2.38845495692263852713418227376, 3.58220034193358197913409908408, 4.55164923330421899238332547875, 4.87065562354408528436738238454, 6.04718050972638017506347344742, 7.28817140568942164865320592150, 7.87161063317590005101558937033, 8.395439613379677401257080046208

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