Properties

Label 2-4025-1.1-c1-0-80
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  + 2.38·2-s − 1.09·3-s + 3.70·4-s − 2.61·6-s − 7-s + 4.05·8-s − 1.79·9-s + 0.391·11-s − 4.05·12-s − 4.90·13-s − 2.38·14-s + 2.29·16-s + 4.88·17-s − 4.29·18-s + 8.50·19-s + 1.09·21-s + 0.935·22-s − 23-s − 4.44·24-s − 11.7·26-s + 5.25·27-s − 3.70·28-s + 4.87·29-s + 6.31·31-s − 2.64·32-s − 0.429·33-s + 11.6·34-s + ⋯
L(s)  = 1  + 1.68·2-s − 0.632·3-s + 1.85·4-s − 1.06·6-s − 0.377·7-s + 1.43·8-s − 0.599·9-s + 0.118·11-s − 1.17·12-s − 1.36·13-s − 0.638·14-s + 0.573·16-s + 1.18·17-s − 1.01·18-s + 1.95·19-s + 0.239·21-s + 0.199·22-s − 0.208·23-s − 0.907·24-s − 2.29·26-s + 1.01·27-s − 0.699·28-s + 0.904·29-s + 1.13·31-s − 0.467·32-s − 0.0747·33-s + 1.99·34-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

\(L(1)\) \(\approx\) \(3.898448156\)
\(L(\frac12)\) \(\approx\) \(3.898448156\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.38T + 2T^{2} \)
3 \( 1 + 1.09T + 3T^{2} \)
11 \( 1 - 0.391T + 11T^{2} \)
13 \( 1 + 4.90T + 13T^{2} \)
17 \( 1 - 4.88T + 17T^{2} \)
19 \( 1 - 8.50T + 19T^{2} \)
29 \( 1 - 4.87T + 29T^{2} \)
31 \( 1 - 6.31T + 31T^{2} \)
37 \( 1 - 8.31T + 37T^{2} \)
41 \( 1 - 3.31T + 41T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 + 1.23T + 47T^{2} \)
53 \( 1 - 7.03T + 53T^{2} \)
59 \( 1 + 9.90T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 - 5.94T + 67T^{2} \)
71 \( 1 + 3.44T + 71T^{2} \)
73 \( 1 - 1.70T + 73T^{2} \)
79 \( 1 - 7.83T + 79T^{2} \)
83 \( 1 - 3.65T + 83T^{2} \)
89 \( 1 - 7.33T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.108529407137645260254337381594, −7.41777344434740541067464733158, −6.64489253125624901474878557296, −5.99542942461168026698641045997, −5.26529925597935629643773174568, −4.96929036072563860609830728180, −3.94318481418851208165038456415, −2.99955569560324949065783972585, −2.59109990806751942198462602042, −0.906143181467533473667367321976, 0.906143181467533473667367321976, 2.59109990806751942198462602042, 2.99955569560324949065783972585, 3.94318481418851208165038456415, 4.96929036072563860609830728180, 5.26529925597935629643773174568, 5.99542942461168026698641045997, 6.64489253125624901474878557296, 7.41777344434740541067464733158, 8.108529407137645260254337381594

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