Properties

Label 2-4025-1.1-c1-0-179
Degree $2$
Conductor $4025$
Sign $1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 7-s + 6·9-s − 11-s − 6·12-s − 7·13-s − 2·14-s − 4·16-s − 3·17-s − 12·18-s − 8·19-s − 3·21-s + 2·22-s − 23-s + 14·26-s − 9·27-s + 2·28-s − 5·29-s − 2·31-s + 8·32-s + 3·33-s + 6·34-s + 12·36-s + 4·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 0.377·7-s + 2·9-s − 0.301·11-s − 1.73·12-s − 1.94·13-s − 0.534·14-s − 16-s − 0.727·17-s − 2.82·18-s − 1.83·19-s − 0.654·21-s + 0.426·22-s − 0.208·23-s + 2.74·26-s − 1.73·27-s + 0.377·28-s − 0.928·29-s − 0.359·31-s + 1.41·32-s + 0.522·33-s + 1.02·34-s + 2·36-s + 0.657·37-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: yes
Arithmetic: yes
Character: $\chi_{4025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + p T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.73303081138440801117624820938, −6.91611918727625257998844270612, −6.58534400370676603036663207089, −5.50228320963604024452011403623, −4.79509968743593084031103946449, −4.27039865695639390758981198822, −2.35192436240687615135038228537, −1.60593211497441081297364910037, 0, 0, 1.60593211497441081297364910037, 2.35192436240687615135038228537, 4.27039865695639390758981198822, 4.79509968743593084031103946449, 5.50228320963604024452011403623, 6.58534400370676603036663207089, 6.91611918727625257998844270612, 7.73303081138440801117624820938

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