Properties

Label 2-5225-1.1-c1-0-128
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  − 1.78·2-s − 0.213·3-s + 1.17·4-s + 0.380·6-s − 3.50·7-s + 1.47·8-s − 2.95·9-s − 11-s − 0.250·12-s + 1.11·13-s + 6.25·14-s − 4.97·16-s + 3.28·17-s + 5.26·18-s + 19-s + 0.749·21-s + 1.78·22-s − 0.303·23-s − 0.314·24-s − 1.98·26-s + 1.27·27-s − 4.11·28-s − 5.23·29-s − 0.126·31-s + 5.90·32-s + 0.213·33-s − 5.84·34-s + ⋯
L(s)  = 1  − 1.25·2-s − 0.123·3-s + 0.585·4-s + 0.155·6-s − 1.32·7-s + 0.521·8-s − 0.984·9-s − 0.301·11-s − 0.0722·12-s + 0.309·13-s + 1.67·14-s − 1.24·16-s + 0.795·17-s + 1.24·18-s + 0.229·19-s + 0.163·21-s + 0.379·22-s − 0.0633·23-s − 0.0642·24-s − 0.390·26-s + 0.244·27-s − 0.777·28-s − 0.972·29-s − 0.0228·31-s + 1.04·32-s + 0.0371·33-s − 1.00·34-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 \( 1 + 1.78T + 2T^{2} \)
3 \( 1 + 0.213T + 3T^{2} \)
7 \( 1 + 3.50T + 7T^{2} \)
13 \( 1 - 1.11T + 13T^{2} \)
17 \( 1 - 3.28T + 17T^{2} \)
23 \( 1 + 0.303T + 23T^{2} \)
29 \( 1 + 5.23T + 29T^{2} \)
31 \( 1 + 0.126T + 31T^{2} \)
37 \( 1 - 5.03T + 37T^{2} \)
41 \( 1 - 0.293T + 41T^{2} \)
43 \( 1 + 0.180T + 43T^{2} \)
47 \( 1 - 3.64T + 47T^{2} \)
53 \( 1 - 1.17T + 53T^{2} \)
59 \( 1 - 5.73T + 59T^{2} \)
61 \( 1 + 3.61T + 61T^{2} \)
67 \( 1 - 5.70T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 2.32T + 79T^{2} \)
83 \( 1 - 2.66T + 83T^{2} \)
89 \( 1 - 0.0652T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−7.907757992467849624885856235133, −7.38329679175263615258083613702, −6.50102103123447261874135444957, −5.87758851616122899027903419614, −5.11230989831120979222090964622, −3.91024516131134218855572898726, −3.13710736735832182035996134358, −2.25503833281576356315239208371, −0.929033178137951673182999170680, 0, 0.929033178137951673182999170680, 2.25503833281576356315239208371, 3.13710736735832182035996134358, 3.91024516131134218855572898726, 5.11230989831120979222090964622, 5.87758851616122899027903419614, 6.50102103123447261874135444957, 7.38329679175263615258083613702, 7.907757992467849624885856235133

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