Properties

Label 2-55e2-1.1-c1-0-14
Degree $2$
Conductor $3025$
Sign $1$
Analytic cond. $24.1547$
Root an. cond. $4.91474$
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  + 0.618·2-s + 0.381·3-s − 1.61·4-s + 0.236·6-s − 3.85·7-s − 2.23·8-s − 2.85·9-s − 0.618·12-s − 1.76·13-s − 2.38·14-s + 1.85·16-s − 1.61·17-s − 1.76·18-s − 6.70·19-s − 1.47·21-s + 7.09·23-s − 0.854·24-s − 1.09·26-s − 2.23·27-s + 6.23·28-s + 3.61·29-s − 3·31-s + 5.61·32-s − 1.00·34-s + 4.61·36-s + 5.76·37-s − 4.14·38-s + ⋯
L(s)  = 1  + 0.437·2-s + 0.220·3-s − 0.809·4-s + 0.0963·6-s − 1.45·7-s − 0.790·8-s − 0.951·9-s − 0.178·12-s − 0.489·13-s − 0.636·14-s + 0.463·16-s − 0.392·17-s − 0.415·18-s − 1.53·19-s − 0.321·21-s + 1.47·23-s − 0.174·24-s − 0.213·26-s − 0.430·27-s + 1.17·28-s + 0.671·29-s − 0.538·31-s + 0.993·32-s − 0.171·34-s + 0.769·36-s + 0.947·37-s − 0.672·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(24.1547\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8416813393\)
\(L(\frac12)\) \(\approx\) \(0.8416813393\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 - 0.618T + 2T^{2} \)
3 \( 1 - 0.381T + 3T^{2} \)
7 \( 1 + 3.85T + 7T^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 + 1.61T + 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 - 7.09T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 5.94T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 - 9.47T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 0.854T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + 0.618T + 97T^{2} \)
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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

−8.827885907428486798157694586701, −8.201446986708997187654314901168, −7.04122266132773873598011453079, −6.32657147907866032854651657737, −5.70333016856872046537591203535, −4.77824801031410966214136859996, −3.97789620170783798226481701237, −3.10436538494088498147198008226, −2.51347707589512151158421849282, −0.48814740112594634628912402620, 0.48814740112594634628912402620, 2.51347707589512151158421849282, 3.10436538494088498147198008226, 3.97789620170783798226481701237, 4.77824801031410966214136859996, 5.70333016856872046537591203535, 6.32657147907866032854651657737, 7.04122266132773873598011453079, 8.201446986708997187654314901168, 8.827885907428486798157694586701

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