Properties

Label 2-35e2-1.1-c3-0-144
Degree $2$
Conductor $1225$
Sign $-1$
Analytic cond. $72.2773$
Root an. cond. $8.50160$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 7·3-s − 4·4-s − 14·6-s + 24·8-s + 22·9-s − 5·11-s − 28·12-s − 14·13-s − 16·16-s − 21·17-s − 44·18-s − 49·19-s + 10·22-s + 159·23-s + 168·24-s + 28·26-s − 35·27-s + 58·29-s − 147·31-s − 160·32-s − 35·33-s + 42·34-s − 88·36-s − 219·37-s + 98·38-s − 98·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·3-s − 1/2·4-s − 0.952·6-s + 1.06·8-s + 0.814·9-s − 0.137·11-s − 0.673·12-s − 0.298·13-s − 1/4·16-s − 0.299·17-s − 0.576·18-s − 0.591·19-s + 0.0969·22-s + 1.44·23-s + 1.42·24-s + 0.211·26-s − 0.249·27-s + 0.371·29-s − 0.851·31-s − 0.883·32-s − 0.184·33-s + 0.211·34-s − 0.407·36-s − 0.973·37-s + 0.418·38-s − 0.402·39-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: yes
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 \( 1 + p T + p^{3} T^{2} \)
3 \( 1 - 7 T + p^{3} T^{2} \)
11 \( 1 + 5 T + p^{3} T^{2} \)
13 \( 1 + 14 T + p^{3} T^{2} \)
17 \( 1 + 21 T + p^{3} T^{2} \)
19 \( 1 + 49 T + p^{3} T^{2} \)
23 \( 1 - 159 T + p^{3} T^{2} \)
29 \( 1 - 2 p T + p^{3} T^{2} \)
31 \( 1 + 147 T + p^{3} T^{2} \)
37 \( 1 + 219 T + p^{3} T^{2} \)
41 \( 1 + 350 T + p^{3} T^{2} \)
43 \( 1 - 124 T + p^{3} T^{2} \)
47 \( 1 - 525 T + p^{3} T^{2} \)
53 \( 1 + 303 T + p^{3} T^{2} \)
59 \( 1 - 105 T + p^{3} T^{2} \)
61 \( 1 - 413 T + p^{3} T^{2} \)
67 \( 1 + 415 T + p^{3} T^{2} \)
71 \( 1 + 432 T + p^{3} T^{2} \)
73 \( 1 + 1113 T + p^{3} T^{2} \)
79 \( 1 + 103 T + p^{3} T^{2} \)
83 \( 1 - 1092 T + p^{3} T^{2} \)
89 \( 1 - 329 T + p^{3} T^{2} \)
97 \( 1 + 882 T + p^{3} T^{2} \)
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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

−8.886327711522927851923459500575, −8.422929433019687427262991787662, −7.55166073790138457375449612935, −6.91053968763467222967525991761, −5.41135651976865722112866791777, −4.45485570286268559131319427293, −3.52657688279674792733660079734, −2.50677885181864927602686466206, −1.43449752759603936256820943859, 0, 1.43449752759603936256820943859, 2.50677885181864927602686466206, 3.52657688279674792733660079734, 4.45485570286268559131319427293, 5.41135651976865722112866791777, 6.91053968763467222967525991761, 7.55166073790138457375449612935, 8.422929433019687427262991787662, 8.886327711522927851923459500575

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