Properties

Label 2-95e2-1.1-c1-0-132
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  + 0.317·2-s − 2.04·3-s − 1.89·4-s − 0.648·6-s − 3.69·7-s − 1.23·8-s + 1.17·9-s − 5.01·11-s + 3.88·12-s − 4.59·13-s − 1.17·14-s + 3.40·16-s + 3.21·17-s + 0.372·18-s + 7.55·21-s − 1.58·22-s − 0.220·23-s + 2.52·24-s − 1.45·26-s + 3.73·27-s + 7.02·28-s − 3.42·29-s − 7.89·31-s + 3.55·32-s + 10.2·33-s + 1.01·34-s − 2.22·36-s + ⋯
L(s)  = 1  + 0.224·2-s − 1.17·3-s − 0.949·4-s − 0.264·6-s − 1.39·7-s − 0.437·8-s + 0.391·9-s − 1.51·11-s + 1.12·12-s − 1.27·13-s − 0.313·14-s + 0.851·16-s + 0.778·17-s + 0.0877·18-s + 1.64·21-s − 0.338·22-s − 0.0460·23-s + 0.515·24-s − 0.285·26-s + 0.717·27-s + 1.32·28-s − 0.635·29-s − 1.41·31-s + 0.628·32-s + 1.78·33-s + 0.174·34-s − 0.371·36-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 \( 1 - 0.317T + 2T^{2} \)
3 \( 1 + 2.04T + 3T^{2} \)
7 \( 1 + 3.69T + 7T^{2} \)
11 \( 1 + 5.01T + 11T^{2} \)
13 \( 1 + 4.59T + 13T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
23 \( 1 + 0.220T + 23T^{2} \)
29 \( 1 + 3.42T + 29T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 - 2.98T + 37T^{2} \)
41 \( 1 - 0.183T + 41T^{2} \)
43 \( 1 + 7.30T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 1.93T + 53T^{2} \)
59 \( 1 - 6.15T + 59T^{2} \)
61 \( 1 - 5.02T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 0.851T + 71T^{2} \)
73 \( 1 - 5.15T + 73T^{2} \)
79 \( 1 + 9.19T + 79T^{2} \)
83 \( 1 - 4.53T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 1.36T + 97T^{2} \)
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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

−7.35197586205642700136471883238, −6.55647866285935433336449568180, −5.74663342373665854181969730787, −5.35188319206979945786701345511, −4.93007037232862740351071710484, −3.89730820080445140438554112505, −3.17665258839032082620029098047, −2.37073487437671883030056792898, −0.62645511976521079566177151506, 0, 0.62645511976521079566177151506, 2.37073487437671883030056792898, 3.17665258839032082620029098047, 3.89730820080445140438554112505, 4.93007037232862740351071710484, 5.35188319206979945786701345511, 5.74663342373665854181969730787, 6.55647866285935433336449568180, 7.35197586205642700136471883238

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