Properties

Label 2-4025-1.1-c1-0-166
Degree $2$
Conductor $4025$
Sign $1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
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  + 2.20·2-s + 2.83·3-s + 2.85·4-s + 6.25·6-s − 7-s + 1.87·8-s + 5.05·9-s + 4.31·11-s + 8.09·12-s + 1.72·13-s − 2.20·14-s − 1.57·16-s − 2.56·17-s + 11.1·18-s + 2.10·19-s − 2.83·21-s + 9.49·22-s − 23-s + 5.31·24-s + 3.79·26-s + 5.83·27-s − 2.85·28-s + 10.3·29-s + 1.07·31-s − 7.21·32-s + 12.2·33-s − 5.65·34-s + ⋯
L(s)  = 1  + 1.55·2-s + 1.63·3-s + 1.42·4-s + 2.55·6-s − 0.377·7-s + 0.662·8-s + 1.68·9-s + 1.29·11-s + 2.33·12-s + 0.478·13-s − 0.588·14-s − 0.393·16-s − 0.623·17-s + 2.62·18-s + 0.483·19-s − 0.619·21-s + 2.02·22-s − 0.208·23-s + 1.08·24-s + 0.744·26-s + 1.12·27-s − 0.538·28-s + 1.91·29-s + 0.192·31-s − 1.27·32-s + 2.12·33-s − 0.970·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(32.1397\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.676789541\)
\(L(\frac12)\) \(\approx\) \(8.676789541\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 - 2.20T + 2T^{2} \)
3 \( 1 - 2.83T + 3T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 - 1.72T + 13T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 - 2.10T + 19T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 8.45T + 37T^{2} \)
41 \( 1 - 1.65T + 41T^{2} \)
43 \( 1 + 8.34T + 43T^{2} \)
47 \( 1 + 4.13T + 47T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 - 5.12T + 59T^{2} \)
61 \( 1 - 5.11T + 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 - 9.34T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 1.72T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + 5.63T + 89T^{2} \)
97 \( 1 - 7.16T + 97T^{2} \)
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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

−8.667576372317212298304515224552, −7.59755859955969650532367300795, −6.61087118070752158228106938065, −6.49527835880037680663518459970, −5.22348386480757888234150693000, −4.36403930567142482048706586536, −3.74737713695459266939802634193, −3.21699385278388959696363512012, −2.45896998835737329001706443057, −1.47249139713243530709525069004, 1.47249139713243530709525069004, 2.45896998835737329001706443057, 3.21699385278388959696363512012, 3.74737713695459266939802634193, 4.36403930567142482048706586536, 5.22348386480757888234150693000, 6.49527835880037680663518459970, 6.61087118070752158228106938065, 7.59755859955969650532367300795, 8.667576372317212298304515224552

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