Properties

Label 2-4025-1.1-c1-0-173
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.55·2-s + 2.51·3-s + 4.55·4-s + 6.44·6-s − 7-s + 6.53·8-s + 3.34·9-s − 1.58·11-s + 11.4·12-s + 4.40·13-s − 2.55·14-s + 7.61·16-s − 1.68·17-s + 8.55·18-s + 0.973·19-s − 2.51·21-s − 4.05·22-s − 23-s + 16.4·24-s + 11.2·26-s + 0.866·27-s − 4.55·28-s + 4.24·29-s − 3.16·31-s + 6.43·32-s − 3.99·33-s − 4.31·34-s + ⋯
L(s)  = 1  + 1.81·2-s + 1.45·3-s + 2.27·4-s + 2.63·6-s − 0.377·7-s + 2.30·8-s + 1.11·9-s − 0.477·11-s + 3.30·12-s + 1.22·13-s − 0.684·14-s + 1.90·16-s − 0.408·17-s + 2.01·18-s + 0.223·19-s − 0.549·21-s − 0.864·22-s − 0.208·23-s + 3.35·24-s + 2.21·26-s + 0.166·27-s − 0.860·28-s + 0.787·29-s − 0.567·31-s + 1.13·32-s − 0.694·33-s − 0.739·34-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

\(L(1)\) \(\approx\) \(9.847791293\)
\(L(\frac12)\) \(\approx\) \(9.847791293\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.55T + 2T^{2} \)
3 \( 1 - 2.51T + 3T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 - 4.40T + 13T^{2} \)
17 \( 1 + 1.68T + 17T^{2} \)
19 \( 1 - 0.973T + 19T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 - 2.80T + 37T^{2} \)
41 \( 1 - 4.98T + 41T^{2} \)
43 \( 1 - 7.98T + 43T^{2} \)
47 \( 1 - 0.837T + 47T^{2} \)
53 \( 1 + 8.86T + 53T^{2} \)
59 \( 1 + 0.234T + 59T^{2} \)
61 \( 1 - 1.49T + 61T^{2} \)
67 \( 1 - 6.22T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 1.29T + 73T^{2} \)
79 \( 1 - 0.866T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.279602353878201166757342179415, −7.60834856853077193737466110118, −6.83768590555126497290333736638, −6.08381288322677762936199915801, −5.42137448967309936840187319567, −4.28817678488288459739631328628, −3.90573471725189725041595117709, −2.98966824350234243196603779610, −2.61895468148576573166423125326, −1.54793209058215148113475726279, 1.54793209058215148113475726279, 2.61895468148576573166423125326, 2.98966824350234243196603779610, 3.90573471725189725041595117709, 4.28817678488288459739631328628, 5.42137448967309936840187319567, 6.08381288322677762936199915801, 6.83768590555126497290333736638, 7.60834856853077193737466110118, 8.279602353878201166757342179415

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