Properties

Label 2-4025-1.1-c1-0-152
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  + 1.23·2-s − 0.767·3-s − 0.485·4-s − 0.944·6-s − 7-s − 3.05·8-s − 2.41·9-s + 2.89·11-s + 0.372·12-s + 5.98·13-s − 1.23·14-s − 2.79·16-s − 3.74·17-s − 2.96·18-s + 3.96·19-s + 0.767·21-s + 3.55·22-s + 23-s + 2.34·24-s + 7.37·26-s + 4.15·27-s + 0.485·28-s − 5.11·29-s − 2.68·31-s + 2.67·32-s − 2.21·33-s − 4.60·34-s + ⋯
L(s)  = 1  + 0.870·2-s − 0.443·3-s − 0.242·4-s − 0.385·6-s − 0.377·7-s − 1.08·8-s − 0.803·9-s + 0.871·11-s + 0.107·12-s + 1.66·13-s − 0.328·14-s − 0.698·16-s − 0.908·17-s − 0.699·18-s + 0.910·19-s + 0.167·21-s + 0.758·22-s + 0.208·23-s + 0.479·24-s + 1.44·26-s + 0.799·27-s + 0.0916·28-s − 0.950·29-s − 0.483·31-s + 0.473·32-s − 0.386·33-s − 0.790·34-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 \( 1 - 1.23T + 2T^{2} \)
3 \( 1 + 0.767T + 3T^{2} \)
11 \( 1 - 2.89T + 11T^{2} \)
13 \( 1 - 5.98T + 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 - 3.96T + 19T^{2} \)
29 \( 1 + 5.11T + 29T^{2} \)
31 \( 1 + 2.68T + 31T^{2} \)
37 \( 1 - 0.781T + 37T^{2} \)
41 \( 1 + 3.75T + 41T^{2} \)
43 \( 1 + 9.64T + 43T^{2} \)
47 \( 1 - 6.15T + 47T^{2} \)
53 \( 1 + 2.86T + 53T^{2} \)
59 \( 1 - 5.12T + 59T^{2} \)
61 \( 1 + 1.72T + 61T^{2} \)
67 \( 1 - 6.58T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 3.72T + 79T^{2} \)
83 \( 1 - 0.558T + 83T^{2} \)
89 \( 1 + 8.00T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−8.261134104762694171375434182664, −6.99233475018946855474435744972, −6.35212557233805156885416979581, −5.77680053857260054576324620395, −5.19986796118429894291897944935, −4.16712216394016571166092570466, −3.60259825887451491997797554716, −2.86029060047020706940276872555, −1.36397137525893084271601071125, 0, 1.36397137525893084271601071125, 2.86029060047020706940276872555, 3.60259825887451491997797554716, 4.16712216394016571166092570466, 5.19986796118429894291897944935, 5.77680053857260054576324620395, 6.35212557233805156885416979581, 6.99233475018946855474435744972, 8.261134104762694171375434182664

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