Properties

Label 2-5225-1.1-c1-0-104
Degree $2$
Conductor $5225$
Sign $-1$
Analytic cond. $41.7218$
Root an. cond. $6.45924$
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  − 0.692·2-s − 1.14·3-s − 1.52·4-s + 0.794·6-s − 3.60·7-s + 2.43·8-s − 1.68·9-s + 11-s + 1.74·12-s − 7.02·13-s + 2.49·14-s + 1.35·16-s − 0.385·17-s + 1.16·18-s + 19-s + 4.13·21-s − 0.692·22-s + 4.88·23-s − 2.79·24-s + 4.86·26-s + 5.37·27-s + 5.47·28-s − 3.20·29-s + 4.99·31-s − 5.81·32-s − 1.14·33-s + 0.266·34-s + ⋯
L(s)  = 1  − 0.489·2-s − 0.662·3-s − 0.760·4-s + 0.324·6-s − 1.36·7-s + 0.862·8-s − 0.561·9-s + 0.301·11-s + 0.503·12-s − 1.94·13-s + 0.667·14-s + 0.337·16-s − 0.0934·17-s + 0.274·18-s + 0.229·19-s + 0.902·21-s − 0.147·22-s + 1.01·23-s − 0.570·24-s + 0.954·26-s + 1.03·27-s + 1.03·28-s − 0.595·29-s + 0.897·31-s − 1.02·32-s − 0.199·33-s + 0.0457·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(41.7218\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + 0.692T + 2T^{2} \)
3 \( 1 + 1.14T + 3T^{2} \)
7 \( 1 + 3.60T + 7T^{2} \)
13 \( 1 + 7.02T + 13T^{2} \)
17 \( 1 + 0.385T + 17T^{2} \)
23 \( 1 - 4.88T + 23T^{2} \)
29 \( 1 + 3.20T + 29T^{2} \)
31 \( 1 - 4.99T + 31T^{2} \)
37 \( 1 - 0.661T + 37T^{2} \)
41 \( 1 - 3.92T + 41T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 - 2.20T + 47T^{2} \)
53 \( 1 + 4.15T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 7.70T + 61T^{2} \)
67 \( 1 - 1.12T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 5.81T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 6.08T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.83861332766746750176689400414, −7.07561204402911952957875899201, −6.53711937695437644312270936604, −5.55370045271507374499023660756, −5.05765832549648599555680290893, −4.23654815657466615547328120168, −3.22965865177050392641167696116, −2.43964649137298870240477611787, −0.812612435635589281403877608434, 0, 0.812612435635589281403877608434, 2.43964649137298870240477611787, 3.22965865177050392641167696116, 4.23654815657466615547328120168, 5.05765832549648599555680290893, 5.55370045271507374499023660756, 6.53711937695437644312270936604, 7.07561204402911952957875899201, 7.83861332766746750176689400414

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