Properties

Label 2-70e2-1.1-c1-0-37
Degree $2$
Conductor $4900$
Sign $-1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.20·3-s + 7.26·9-s + 4.20·11-s + 0.204·13-s − 5.06·17-s + 1.06·19-s + 2.14·23-s − 13.6·27-s − 7.47·29-s − 8.47·31-s − 13.4·33-s − 10.6·37-s − 0.654·39-s + 10.5·41-s + 8.26·43-s + 3.26·47-s + 16.2·51-s + 5.67·53-s − 3.40·57-s + 1.20·59-s + 1.65·61-s − 12.4·67-s − 6.85·69-s − 0.591·71-s − 4·73-s + 6.54·79-s + 22.0·81-s + ⋯
L(s)  = 1  − 1.85·3-s + 2.42·9-s + 1.26·11-s + 0.0566·13-s − 1.22·17-s + 0.244·19-s + 0.446·23-s − 2.63·27-s − 1.38·29-s − 1.52·31-s − 2.34·33-s − 1.74·37-s − 0.104·39-s + 1.64·41-s + 1.26·43-s + 0.476·47-s + 2.27·51-s + 0.779·53-s − 0.451·57-s + 0.156·59-s + 0.211·61-s − 1.51·67-s − 0.825·69-s − 0.0701·71-s − 0.468·73-s + 0.736·79-s + 2.44·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 3.20T + 3T^{2} \)
11 \( 1 - 4.20T + 11T^{2} \)
13 \( 1 - 0.204T + 13T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 - 1.06T + 19T^{2} \)
23 \( 1 - 2.14T + 23T^{2} \)
29 \( 1 + 7.47T + 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 8.26T + 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 - 5.67T + 53T^{2} \)
59 \( 1 - 1.20T + 59T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 0.591T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 6.54T + 79T^{2} \)
83 \( 1 - 3.88T + 83T^{2} \)
89 \( 1 + 9.26T + 89T^{2} \)
97 \( 1 - 1.33T + 97T^{2} \)
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

−7.38326589098707973131463622273, −7.15933245211294192484394288746, −6.30564278188283802954254347446, −5.79320526326866992594591918765, −5.10519408038735440619761125044, −4.24805233955196192141229409764, −3.72611543418915920743764371028, −2.05180450179778004754934583423, −1.13338499387032574266983770810, 0, 1.13338499387032574266983770810, 2.05180450179778004754934583423, 3.72611543418915920743764371028, 4.24805233955196192141229409764, 5.10519408038735440619761125044, 5.79320526326866992594591918765, 6.30564278188283802954254347446, 7.15933245211294192484394288746, 7.38326589098707973131463622273

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