Properties

Label 2-35e2-1.1-c3-0-150
Degree $2$
Conductor $1225$
Sign $-1$
Analytic cond. $72.2773$
Root an. cond. $8.50160$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.58·2-s + 6.65·3-s − 1.31·4-s − 17.2·6-s + 24.0·8-s + 17.3·9-s + 38.2·11-s − 8.74·12-s + 19.3·13-s − 51.7·16-s − 87.2·17-s − 44.7·18-s + 44.2·19-s − 98.9·22-s − 218.·23-s + 160.·24-s − 50.0·26-s − 64.4·27-s − 46.9·29-s − 194.·31-s − 58.8·32-s + 254.·33-s + 225.·34-s − 22.7·36-s − 366.·37-s − 114.·38-s + 128.·39-s + ⋯
L(s)  = 1  − 0.914·2-s + 1.28·3-s − 0.164·4-s − 1.17·6-s + 1.06·8-s + 0.641·9-s + 1.04·11-s − 0.210·12-s + 0.412·13-s − 0.808·16-s − 1.24·17-s − 0.586·18-s + 0.534·19-s − 0.958·22-s − 1.97·23-s + 1.36·24-s − 0.377·26-s − 0.459·27-s − 0.300·29-s − 1.12·31-s − 0.324·32-s + 1.34·33-s + 1.13·34-s − 0.105·36-s − 1.63·37-s − 0.488·38-s + 0.528·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(72.2773\)
Root analytic conductor: \(8.50160\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + 2.58T + 8T^{2} \)
3 \( 1 - 6.65T + 27T^{2} \)
11 \( 1 - 38.2T + 1.33e3T^{2} \)
13 \( 1 - 19.3T + 2.19e3T^{2} \)
17 \( 1 + 87.2T + 4.91e3T^{2} \)
19 \( 1 - 44.2T + 6.85e3T^{2} \)
23 \( 1 + 218.T + 1.21e4T^{2} \)
29 \( 1 + 46.9T + 2.43e4T^{2} \)
31 \( 1 + 194.T + 2.97e4T^{2} \)
37 \( 1 + 366.T + 5.06e4T^{2} \)
41 \( 1 - 339.T + 6.89e4T^{2} \)
43 \( 1 - 226.T + 7.95e4T^{2} \)
47 \( 1 - 11.6T + 1.03e5T^{2} \)
53 \( 1 - 209.T + 1.48e5T^{2} \)
59 \( 1 - 616T + 2.05e5T^{2} \)
61 \( 1 + 320.T + 2.26e5T^{2} \)
67 \( 1 + 14.5T + 3.00e5T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 - 824.T + 3.89e5T^{2} \)
79 \( 1 - 156.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 - 170.T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3T + 9.12e5T^{2} \)
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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

−9.063735970955052681629463298847, −8.342058649674969757797721330066, −7.64428176531600296330580043408, −6.82529093893812924891762575321, −5.63948132721623645014725336614, −4.16956198183376017704801919054, −3.76025736354589034692134286515, −2.29693089291660890722972924719, −1.47822155308891159926625694764, 0, 1.47822155308891159926625694764, 2.29693089291660890722972924719, 3.76025736354589034692134286515, 4.16956198183376017704801919054, 5.63948132721623645014725336614, 6.82529093893812924891762575321, 7.64428176531600296330580043408, 8.342058649674969757797721330066, 9.063735970955052681629463298847

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