Properties

Label 2-95e2-1.1-c1-0-227
Degree $2$
Conductor $9025$
Sign $-1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·2-s − 2.23·3-s − 1.85·4-s + 0.854·6-s + 0.236·7-s + 1.47·8-s + 2.00·9-s − 3.47·11-s + 4.14·12-s + 2·13-s − 0.0901·14-s + 3.14·16-s + 1.85·17-s − 0.763·18-s − 0.527·21-s + 1.32·22-s + 3.23·23-s − 3.29·24-s − 0.763·26-s + 2.23·27-s − 0.437·28-s − 6·29-s − 4.85·31-s − 4.14·32-s + 7.76·33-s − 0.708·34-s − 3.70·36-s + ⋯
L(s)  = 1  − 0.270·2-s − 1.29·3-s − 0.927·4-s + 0.348·6-s + 0.0892·7-s + 0.520·8-s + 0.666·9-s − 1.04·11-s + 1.19·12-s + 0.554·13-s − 0.0240·14-s + 0.786·16-s + 0.449·17-s − 0.180·18-s − 0.115·21-s + 0.282·22-s + 0.674·23-s − 0.671·24-s − 0.149·26-s + 0.430·27-s − 0.0827·28-s − 1.11·29-s − 0.871·31-s − 0.732·32-s + 1.35·33-s − 0.121·34-s − 0.618·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + 0.381T + 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 - 0.236T + 7T^{2} \)
11 \( 1 + 3.47T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4.85T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 - 8.61T + 41T^{2} \)
43 \( 1 + 9.56T + 43T^{2} \)
47 \( 1 + 5.38T + 47T^{2} \)
53 \( 1 - 8.61T + 53T^{2} \)
59 \( 1 - 7.23T + 59T^{2} \)
61 \( 1 - 14.5T + 61T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 9.70T + 89T^{2} \)
97 \( 1 + 9.61T + 97T^{2} \)
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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

−7.35531475183348738192944558297, −6.71619083831045008883991278067, −5.75132689134770131040437761502, −5.28038611522135665237498538858, −4.97546596505039016273070379263, −3.93287738711975059855197663193, −3.25956834176814009342240549881, −1.90307919478012105254631857328, −0.845619292405495196405407857337, 0, 0.845619292405495196405407857337, 1.90307919478012105254631857328, 3.25956834176814009342240549881, 3.93287738711975059855197663193, 4.97546596505039016273070379263, 5.28038611522135665237498538858, 5.75132689134770131040437761502, 6.71619083831045008883991278067, 7.35531475183348738192944558297

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