Properties

Label 2-6025-1.1-c1-0-68
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  − 1.77·2-s − 0.796·3-s + 1.14·4-s + 1.41·6-s − 3.30·7-s + 1.52·8-s − 2.36·9-s + 2.56·11-s − 0.909·12-s + 4.28·13-s + 5.86·14-s − 4.97·16-s + 3.96·17-s + 4.19·18-s − 3.96·19-s + 2.63·21-s − 4.54·22-s + 2.74·23-s − 1.21·24-s − 7.59·26-s + 4.27·27-s − 3.77·28-s − 3.80·29-s + 4.48·31-s + 5.78·32-s − 2.04·33-s − 7.02·34-s + ⋯
L(s)  = 1  − 1.25·2-s − 0.459·3-s + 0.571·4-s + 0.576·6-s − 1.24·7-s + 0.537·8-s − 0.788·9-s + 0.773·11-s − 0.262·12-s + 1.18·13-s + 1.56·14-s − 1.24·16-s + 0.961·17-s + 0.988·18-s − 0.908·19-s + 0.574·21-s − 0.969·22-s + 0.573·23-s − 0.247·24-s − 1.48·26-s + 0.822·27-s − 0.713·28-s − 0.707·29-s + 0.805·31-s + 1.02·32-s − 0.355·33-s − 1.20·34-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

\(L(1)\) \(\approx\) \(0.5673649643\)
\(L(\frac12)\) \(\approx\) \(0.5673649643\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.77T + 2T^{2} \)
3 \( 1 + 0.796T + 3T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 - 4.28T + 13T^{2} \)
17 \( 1 - 3.96T + 17T^{2} \)
19 \( 1 + 3.96T + 19T^{2} \)
23 \( 1 - 2.74T + 23T^{2} \)
29 \( 1 + 3.80T + 29T^{2} \)
31 \( 1 - 4.48T + 31T^{2} \)
37 \( 1 + 1.87T + 37T^{2} \)
41 \( 1 - 1.91T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + 1.93T + 47T^{2} \)
53 \( 1 - 1.47T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 2.78T + 61T^{2} \)
67 \( 1 - 4.70T + 67T^{2} \)
71 \( 1 - 0.780T + 71T^{2} \)
73 \( 1 - 1.79T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 9.66T + 97T^{2} \)
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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

−8.281402651138387253401577576657, −7.48231134140212529450981191518, −6.61062224319043208627603890422, −6.21315586837397410520809197185, −5.50041702231531852173309925776, −4.33132389324329134125026785694, −3.55675423768603757876453000793, −2.68042068680610247606807946390, −1.38401754417647204446876700412, −0.53675340180908903006478963772, 0.53675340180908903006478963772, 1.38401754417647204446876700412, 2.68042068680610247606807946390, 3.55675423768603757876453000793, 4.33132389324329134125026785694, 5.50041702231531852173309925776, 6.21315586837397410520809197185, 6.61062224319043208627603890422, 7.48231134140212529450981191518, 8.281402651138387253401577576657

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