Properties

Label 2-55e2-1.1-c1-0-137
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  + 2.56·2-s + 1.79·3-s + 4.56·4-s + 4.59·6-s + 3.80·7-s + 6.57·8-s + 0.213·9-s + 8.18·12-s − 4.49·13-s + 9.73·14-s + 7.71·16-s + 5.01·17-s + 0.546·18-s − 3.46·19-s + 6.81·21-s + 0.105·23-s + 11.7·24-s − 11.5·26-s − 4.99·27-s + 17.3·28-s − 2.35·29-s + 4.74·31-s + 6.62·32-s + 12.8·34-s + 0.974·36-s − 5.60·37-s − 8.86·38-s + ⋯
L(s)  = 1  + 1.81·2-s + 1.03·3-s + 2.28·4-s + 1.87·6-s + 1.43·7-s + 2.32·8-s + 0.0711·9-s + 2.36·12-s − 1.24·13-s + 2.60·14-s + 1.92·16-s + 1.21·17-s + 0.128·18-s − 0.793·19-s + 1.48·21-s + 0.0220·23-s + 2.40·24-s − 2.26·26-s − 0.961·27-s + 3.27·28-s − 0.436·29-s + 0.852·31-s + 1.17·32-s + 2.20·34-s + 0.162·36-s − 0.920·37-s − 1.43·38-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.863714600\)
\(L(\frac12)\) \(\approx\) \(8.863714600\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 2T^{2} \)
3 \( 1 - 1.79T + 3T^{2} \)
7 \( 1 - 3.80T + 7T^{2} \)
13 \( 1 + 4.49T + 13T^{2} \)
17 \( 1 - 5.01T + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 - 0.105T + 23T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 + 5.60T + 37T^{2} \)
41 \( 1 - 0.916T + 41T^{2} \)
43 \( 1 + 4.46T + 43T^{2} \)
47 \( 1 - 1.25T + 47T^{2} \)
53 \( 1 + 5.49T + 53T^{2} \)
59 \( 1 + 1.18T + 59T^{2} \)
61 \( 1 + 7.32T + 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + 2.18T + 71T^{2} \)
73 \( 1 - 8.95T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 4.71T + 83T^{2} \)
89 \( 1 + 0.172T + 89T^{2} \)
97 \( 1 + 4.46T + 97T^{2} \)
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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

−8.383064726851346365536932811799, −7.83136376566108758766397130219, −7.24955699921671542523434079250, −6.24563701523679692079262755761, −5.27346456027390480705618358518, −4.88146051534187280322077360947, −4.01491377667170165795874658276, −3.20118767049142718885447908731, −2.39320774559316375551510409920, −1.71120679492796465412732604213, 1.71120679492796465412732604213, 2.39320774559316375551510409920, 3.20118767049142718885447908731, 4.01491377667170165795874658276, 4.88146051534187280322077360947, 5.27346456027390480705618358518, 6.24563701523679692079262755761, 7.24955699921671542523434079250, 7.83136376566108758766397130219, 8.383064726851346365536932811799

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