Properties

Label 2-6025-1.1-c1-0-145
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  − 0.428·2-s − 2.33·3-s − 1.81·4-s + 0.998·6-s + 3.85·7-s + 1.63·8-s + 2.43·9-s − 0.897·11-s + 4.23·12-s + 4.05·13-s − 1.65·14-s + 2.93·16-s + 3.01·17-s − 1.04·18-s − 5.78·19-s − 8.99·21-s + 0.384·22-s + 7.46·23-s − 3.81·24-s − 1.73·26-s + 1.31·27-s − 7.00·28-s + 9.45·29-s + 8.89·31-s − 4.52·32-s + 2.09·33-s − 1.29·34-s + ⋯
L(s)  = 1  − 0.302·2-s − 1.34·3-s − 0.908·4-s + 0.407·6-s + 1.45·7-s + 0.577·8-s + 0.811·9-s − 0.270·11-s + 1.22·12-s + 1.12·13-s − 0.441·14-s + 0.733·16-s + 0.732·17-s − 0.245·18-s − 1.32·19-s − 1.96·21-s + 0.0819·22-s + 1.55·23-s − 0.777·24-s − 0.340·26-s + 0.253·27-s − 1.32·28-s + 1.75·29-s + 1.59·31-s − 0.799·32-s + 0.364·33-s − 0.221·34-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

\(L(1)\) \(\approx\) \(1.231883141\)
\(L(\frac12)\) \(\approx\) \(1.231883141\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.428T + 2T^{2} \)
3 \( 1 + 2.33T + 3T^{2} \)
7 \( 1 - 3.85T + 7T^{2} \)
11 \( 1 + 0.897T + 11T^{2} \)
13 \( 1 - 4.05T + 13T^{2} \)
17 \( 1 - 3.01T + 17T^{2} \)
19 \( 1 + 5.78T + 19T^{2} \)
23 \( 1 - 7.46T + 23T^{2} \)
29 \( 1 - 9.45T + 29T^{2} \)
31 \( 1 - 8.89T + 31T^{2} \)
37 \( 1 - 9.60T + 37T^{2} \)
41 \( 1 - 5.69T + 41T^{2} \)
43 \( 1 + 3.09T + 43T^{2} \)
47 \( 1 - 9.18T + 47T^{2} \)
53 \( 1 + 3.93T + 53T^{2} \)
59 \( 1 + 6.77T + 59T^{2} \)
61 \( 1 + 1.83T + 61T^{2} \)
67 \( 1 - 8.14T + 67T^{2} \)
71 \( 1 - 6.27T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 4.57T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 6.63T + 97T^{2} \)
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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

−8.279388425300818322647631111535, −7.50310229651440589207165804611, −6.41351044320992870502346343175, −5.95796972564006637240263065835, −4.96757734536491427877108169952, −4.76925637777911051807827653940, −4.04352935201219988662632506653, −2.70817880037872178440536602405, −1.19650731321265511077735403147, −0.846438957358517436873463481500, 0.846438957358517436873463481500, 1.19650731321265511077735403147, 2.70817880037872178440536602405, 4.04352935201219988662632506653, 4.76925637777911051807827653940, 4.96757734536491427877108169952, 5.95796972564006637240263065835, 6.41351044320992870502346343175, 7.50310229651440589207165804611, 8.279388425300818322647631111535

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