Properties

Label 2-35e2-1.1-c3-0-76
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  − 5.41·2-s − 4.65·3-s + 21.3·4-s + 25.2·6-s − 72.0·8-s − 5.31·9-s − 52.2·11-s − 99.2·12-s + 30.6·13-s + 219.·16-s + 37.2·17-s + 28.7·18-s − 80.2·19-s + 282.·22-s − 25.8·23-s + 335.·24-s − 165.·26-s + 150.·27-s + 20.9·29-s + 314.·31-s − 613.·32-s + 243.·33-s − 201.·34-s − 113.·36-s − 197.·37-s + 434.·38-s − 142.·39-s + ⋯
L(s)  = 1  − 1.91·2-s − 0.896·3-s + 2.66·4-s + 1.71·6-s − 3.18·8-s − 0.196·9-s − 1.43·11-s − 2.38·12-s + 0.654·13-s + 3.43·16-s + 0.531·17-s + 0.376·18-s − 0.968·19-s + 2.74·22-s − 0.234·23-s + 2.85·24-s − 1.25·26-s + 1.07·27-s + 0.134·29-s + 1.82·31-s − 3.38·32-s + 1.28·33-s − 1.01·34-s − 0.524·36-s − 0.875·37-s + 1.85·38-s − 0.586·39-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 \( 1 + 5.41T + 8T^{2} \)
3 \( 1 + 4.65T + 27T^{2} \)
11 \( 1 + 52.2T + 1.33e3T^{2} \)
13 \( 1 - 30.6T + 2.19e3T^{2} \)
17 \( 1 - 37.2T + 4.91e3T^{2} \)
19 \( 1 + 80.2T + 6.85e3T^{2} \)
23 \( 1 + 25.8T + 1.21e4T^{2} \)
29 \( 1 - 20.9T + 2.43e4T^{2} \)
31 \( 1 - 314.T + 2.97e4T^{2} \)
37 \( 1 + 197.T + 5.06e4T^{2} \)
41 \( 1 + 11.3T + 6.89e4T^{2} \)
43 \( 1 - 33.8T + 7.95e4T^{2} \)
47 \( 1 + 361.T + 1.03e5T^{2} \)
53 \( 1 + 153.T + 1.48e5T^{2} \)
59 \( 1 - 616T + 2.05e5T^{2} \)
61 \( 1 + 15.2T + 2.26e5T^{2} \)
67 \( 1 - 166.T + 3.00e5T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 + 148.T + 3.89e5T^{2} \)
79 \( 1 - 857.T + 4.93e5T^{2} \)
83 \( 1 - 660.T + 5.71e5T^{2} \)
89 \( 1 - 45.7T + 7.04e5T^{2} \)
97 \( 1 - 1.68e3T + 9.12e5T^{2} \)
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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

−8.753121347542266478128498846534, −8.299694689327958777216252248981, −7.56593629459955471466369014225, −6.51287040492505255272973767710, −6.01292167170788064406480949044, −4.98641766015974544172455919274, −3.14854441715333259544207396108, −2.16547646323359179056678802407, −0.872637716312671756744814455202, 0, 0.872637716312671756744814455202, 2.16547646323359179056678802407, 3.14854441715333259544207396108, 4.98641766015974544172455919274, 6.01292167170788064406480949044, 6.51287040492505255272973767710, 7.56593629459955471466369014225, 8.299694689327958777216252248981, 8.753121347542266478128498846534

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