Properties

Label 2-4025-1.1-c1-0-203
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.83·2-s + 1.64·3-s + 1.38·4-s + 3.02·6-s − 7-s − 1.13·8-s − 0.291·9-s − 2.13·11-s + 2.27·12-s − 3.30·13-s − 1.83·14-s − 4.85·16-s − 4.46·17-s − 0.536·18-s + 4.87·19-s − 1.64·21-s − 3.92·22-s + 23-s − 1.86·24-s − 6.08·26-s − 5.41·27-s − 1.38·28-s − 7.83·29-s + 10.2·31-s − 6.65·32-s − 3.51·33-s − 8.22·34-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.950·3-s + 0.691·4-s + 1.23·6-s − 0.377·7-s − 0.400·8-s − 0.0972·9-s − 0.644·11-s + 0.657·12-s − 0.916·13-s − 0.491·14-s − 1.21·16-s − 1.08·17-s − 0.126·18-s + 1.11·19-s − 0.359·21-s − 0.837·22-s + 0.208·23-s − 0.380·24-s − 1.19·26-s − 1.04·27-s − 0.261·28-s − 1.45·29-s + 1.83·31-s − 1.17·32-s − 0.611·33-s − 1.40·34-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: \(1\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good2 \( 1 - 1.83T + 2T^{2} \)
3 \( 1 - 1.64T + 3T^{2} \)
11 \( 1 + 2.13T + 11T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
17 \( 1 + 4.46T + 17T^{2} \)
19 \( 1 - 4.87T + 19T^{2} \)
29 \( 1 + 7.83T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 - 1.83T + 41T^{2} \)
43 \( 1 - 0.176T + 43T^{2} \)
47 \( 1 + 5.16T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + 6.55T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 - 4.64T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 - 8.95T + 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−7.898788030413658213164710128383, −7.39356175483370775162599858998, −6.39519547306054673237249186088, −5.75491045699257099287093122650, −4.85760871556697086067493625393, −4.35018256930571282678933427480, −3.15621217267932336059396455214, −2.96197372037138390344071675788, −2.02987198274690395249469342615, 0, 2.02987198274690395249469342615, 2.96197372037138390344071675788, 3.15621217267932336059396455214, 4.35018256930571282678933427480, 4.85760871556697086067493625393, 5.75491045699257099287093122650, 6.39519547306054673237249186088, 7.39356175483370775162599858998, 7.898788030413658213164710128383

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