Properties

Label 2-6025-1.1-c1-0-97
Degree $2$
Conductor $6025$
Sign $1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.07·2-s + 2.50·3-s − 0.837·4-s + 2.70·6-s − 4.80·7-s − 3.05·8-s + 3.28·9-s − 2.54·11-s − 2.09·12-s − 4.01·13-s − 5.18·14-s − 1.62·16-s + 4.78·17-s + 3.54·18-s + 3.43·19-s − 12.0·21-s − 2.74·22-s + 4.75·23-s − 7.67·24-s − 4.33·26-s + 0.723·27-s + 4.02·28-s + 3.27·29-s + 0.120·31-s + 4.36·32-s − 6.39·33-s + 5.15·34-s + ⋯
L(s)  = 1  + 0.762·2-s + 1.44·3-s − 0.418·4-s + 1.10·6-s − 1.81·7-s − 1.08·8-s + 1.09·9-s − 0.768·11-s − 0.605·12-s − 1.11·13-s − 1.38·14-s − 0.406·16-s + 1.16·17-s + 0.835·18-s + 0.788·19-s − 2.62·21-s − 0.586·22-s + 0.991·23-s − 1.56·24-s − 0.849·26-s + 0.139·27-s + 0.760·28-s + 0.607·29-s + 0.0215·31-s + 0.771·32-s − 1.11·33-s + 0.884·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.753527016\)
\(L(\frac12)\) \(\approx\) \(2.753527016\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 - 1.07T + 2T^{2} \)
3 \( 1 - 2.50T + 3T^{2} \)
7 \( 1 + 4.80T + 7T^{2} \)
11 \( 1 + 2.54T + 11T^{2} \)
13 \( 1 + 4.01T + 13T^{2} \)
17 \( 1 - 4.78T + 17T^{2} \)
19 \( 1 - 3.43T + 19T^{2} \)
23 \( 1 - 4.75T + 23T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 - 0.120T + 31T^{2} \)
37 \( 1 - 5.16T + 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 + 0.519T + 43T^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 - 0.643T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 + 1.34T + 61T^{2} \)
67 \( 1 + 5.79T + 67T^{2} \)
71 \( 1 - 4.79T + 71T^{2} \)
73 \( 1 + 2.89T + 73T^{2} \)
79 \( 1 - 7.95T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 3.69T + 89T^{2} \)
97 \( 1 - 5.96T + 97T^{2} \)
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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

−8.004152598081657051695364677035, −7.44081190172582236748257558713, −6.73333947770534575946163376906, −5.74218419185891136180732815669, −5.21238014931113972183003153442, −4.17233369206523753551374615503, −3.50293706479997303003027066678, −2.74094144437927154057666846245, −2.71272598232545356045124960749, −0.69336700354620242259947555636, 0.69336700354620242259947555636, 2.71272598232545356045124960749, 2.74094144437927154057666846245, 3.50293706479997303003027066678, 4.17233369206523753551374615503, 5.21238014931113972183003153442, 5.74218419185891136180732815669, 6.73333947770534575946163376906, 7.44081190172582236748257558713, 8.004152598081657051695364677035

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