Properties

Label 2-55e2-1.1-c1-0-144
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  + 1.93·2-s − 0.517·3-s + 1.73·4-s − 0.999·6-s + 0.896·7-s − 0.517·8-s − 2.73·9-s − 0.896·12-s − 4.24·13-s + 1.73·14-s − 4.46·16-s + 1.03·17-s − 5.27·18-s + 6.19·19-s − 0.464·21-s + 6.31·23-s + 0.267·24-s − 8.19·26-s + 2.96·27-s + 1.55·28-s − 6.92·29-s − 5.26·31-s − 7.58·32-s + 1.99·34-s − 4.73·36-s − 6.69·37-s + 11.9·38-s + ⋯
L(s)  = 1  + 1.36·2-s − 0.298·3-s + 0.866·4-s − 0.408·6-s + 0.338·7-s − 0.183·8-s − 0.910·9-s − 0.258·12-s − 1.17·13-s + 0.462·14-s − 1.11·16-s + 0.251·17-s − 1.24·18-s + 1.42·19-s − 0.101·21-s + 1.31·23-s + 0.0546·24-s − 1.60·26-s + 0.571·27-s + 0.293·28-s − 1.28·29-s − 0.946·31-s − 1.34·32-s + 0.342·34-s − 0.788·36-s − 1.10·37-s + 1.94·38-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 \( 1 - 1.93T + 2T^{2} \)
3 \( 1 + 0.517T + 3T^{2} \)
7 \( 1 - 0.896T + 7T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 1.03T + 17T^{2} \)
19 \( 1 - 6.19T + 19T^{2} \)
23 \( 1 - 6.31T + 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + 6.69T + 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 4.10T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 4.73T + 59T^{2} \)
61 \( 1 + 8.26T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 + 5.46T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 6.46T + 89T^{2} \)
97 \( 1 + 0.656T + 97T^{2} \)
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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

−8.230923180519535573432058893734, −7.33025995915307542502557435481, −6.69302483911069970942838083984, −5.68295117238408958911270163393, −5.13915486301664176180498898473, −4.79736981394238779451550480042, −3.40408663410559302778914222891, −3.07364467709568210214719778214, −1.81384883113694686248083067958, 0, 1.81384883113694686248083067958, 3.07364467709568210214719778214, 3.40408663410559302778914222891, 4.79736981394238779451550480042, 5.13915486301664176180498898473, 5.68295117238408958911270163393, 6.69302483911069970942838083984, 7.33025995915307542502557435481, 8.230923180519535573432058893734

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