Properties

Label 2-5225-1.1-c1-0-126
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  − 2.40·2-s − 1.94·3-s + 3.78·4-s + 4.68·6-s + 0.665·7-s − 4.28·8-s + 0.794·9-s + 11-s − 7.36·12-s − 3.05·13-s − 1.59·14-s + 2.73·16-s − 4.49·17-s − 1.91·18-s + 19-s − 1.29·21-s − 2.40·22-s − 0.205·23-s + 8.34·24-s + 7.34·26-s + 4.29·27-s + 2.51·28-s + 0.0417·29-s + 6.76·31-s + 1.98·32-s − 1.94·33-s + 10.8·34-s + ⋯
L(s)  = 1  − 1.70·2-s − 1.12·3-s + 1.89·4-s + 1.91·6-s + 0.251·7-s − 1.51·8-s + 0.264·9-s + 0.301·11-s − 2.12·12-s − 0.847·13-s − 0.427·14-s + 0.683·16-s − 1.08·17-s − 0.450·18-s + 0.229·19-s − 0.282·21-s − 0.512·22-s − 0.0427·23-s + 1.70·24-s + 1.44·26-s + 0.826·27-s + 0.475·28-s + 0.00776·29-s + 1.21·31-s + 0.351·32-s − 0.339·33-s + 1.85·34-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 \( 1 + 2.40T + 2T^{2} \)
3 \( 1 + 1.94T + 3T^{2} \)
7 \( 1 - 0.665T + 7T^{2} \)
13 \( 1 + 3.05T + 13T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
23 \( 1 + 0.205T + 23T^{2} \)
29 \( 1 - 0.0417T + 29T^{2} \)
31 \( 1 - 6.76T + 31T^{2} \)
37 \( 1 + 7.60T + 37T^{2} \)
41 \( 1 - 5.90T + 41T^{2} \)
43 \( 1 - 1.77T + 43T^{2} \)
47 \( 1 - 2.44T + 47T^{2} \)
53 \( 1 - 3.34T + 53T^{2} \)
59 \( 1 - 7.44T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 7.55T + 67T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 - 1.57T + 73T^{2} \)
79 \( 1 + 4.66T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 6.06T + 89T^{2} \)
97 \( 1 + 8.44T + 97T^{2} \)
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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

−7.946194217362719797200709723105, −7.13037113621686789514494189487, −6.66558976089474192655477424714, −5.96548819210036455423392333860, −5.05217475371365424042519461348, −4.32857592161366286608648720579, −2.85324066764311997070436277835, −1.98223261037008626286270678481, −0.916856295511262901570801938728, 0, 0.916856295511262901570801938728, 1.98223261037008626286270678481, 2.85324066764311997070436277835, 4.32857592161366286608648720579, 5.05217475371365424042519461348, 5.96548819210036455423392333860, 6.66558976089474192655477424714, 7.13037113621686789514494189487, 7.946194217362719797200709723105

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