Properties

Label 2-4025-1.1-c1-0-30
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  + 0.193·2-s − 2.48·3-s − 1.96·4-s − 0.481·6-s + 7-s − 0.768·8-s + 3.15·9-s + 1.19·11-s + 4.86·12-s + 2.96·13-s + 0.193·14-s + 3.77·16-s + 0.481·17-s + 0.612·18-s − 4.31·19-s − 2.48·21-s + 0.231·22-s − 23-s + 1.90·24-s + 0.574·26-s − 0.387·27-s − 1.96·28-s − 4.15·29-s + 10.2·31-s + 2.26·32-s − 2.96·33-s + 0.0933·34-s + ⋯
L(s)  = 1  + 0.137·2-s − 1.43·3-s − 0.981·4-s − 0.196·6-s + 0.377·7-s − 0.271·8-s + 1.05·9-s + 0.359·11-s + 1.40·12-s + 0.821·13-s + 0.0518·14-s + 0.943·16-s + 0.116·17-s + 0.144·18-s − 0.989·19-s − 0.541·21-s + 0.0493·22-s − 0.208·23-s + 0.389·24-s + 0.112·26-s − 0.0746·27-s − 0.370·28-s − 0.771·29-s + 1.83·31-s + 0.401·32-s − 0.515·33-s + 0.0160·34-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

\(L(1)\) \(\approx\) \(0.7981877048\)
\(L(\frac12)\) \(\approx\) \(0.7981877048\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.193T + 2T^{2} \)
3 \( 1 + 2.48T + 3T^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
17 \( 1 - 0.481T + 17T^{2} \)
19 \( 1 + 4.31T + 19T^{2} \)
29 \( 1 + 4.15T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 5.35T + 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 - 5.73T + 43T^{2} \)
47 \( 1 + 0.168T + 47T^{2} \)
53 \( 1 + 8.70T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 9.05T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 0.775T + 71T^{2} \)
73 \( 1 + 8.88T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 0.0933T + 97T^{2} \)
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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

−8.454311412511253611039110500098, −7.79807169845103615919730102261, −6.63264666279038371027171497431, −6.11418879267833437266980956828, −5.51360607472759548503880074758, −4.61207763373888271781025574341, −4.28171840738164095283746032644, −3.18331514332770643042462514844, −1.58069807411594877123945982277, −0.57473501613142959468924361549, 0.57473501613142959468924361549, 1.58069807411594877123945982277, 3.18331514332770643042462514844, 4.28171840738164095283746032644, 4.61207763373888271781025574341, 5.51360607472759548503880074758, 6.11418879267833437266980956828, 6.63264666279038371027171497431, 7.79807169845103615919730102261, 8.454311412511253611039110500098

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