Properties

Label 2-35e2-1.1-c3-0-178
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  + 5·2-s + 17·4-s + 45·8-s − 27·9-s − 68·11-s + 89·16-s − 135·18-s − 340·22-s + 40·23-s − 166·29-s + 85·32-s − 459·36-s − 450·37-s + 180·43-s − 1.15e3·44-s + 200·46-s − 590·53-s − 830·58-s − 287·64-s + 740·67-s + 688·71-s − 1.21e3·72-s − 2.25e3·74-s − 1.38e3·79-s + 729·81-s + 900·86-s − 3.06e3·88-s + ⋯
L(s)  = 1  + 1.76·2-s + 17/8·4-s + 1.98·8-s − 9-s − 1.86·11-s + 1.39·16-s − 1.76·18-s − 3.29·22-s + 0.362·23-s − 1.06·29-s + 0.469·32-s − 2.12·36-s − 1.99·37-s + 0.638·43-s − 3.96·44-s + 0.641·46-s − 1.52·53-s − 1.87·58-s − 0.560·64-s + 1.34·67-s + 1.15·71-s − 1.98·72-s − 3.53·74-s − 1.97·79-s + 81-s + 1.12·86-s − 3.70·88-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 \( 1 - 5 T + p^{3} T^{2} \)
3 \( 1 + p^{3} T^{2} \)
11 \( 1 + 68 T + p^{3} T^{2} \)
13 \( 1 + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 + p^{3} T^{2} \)
23 \( 1 - 40 T + p^{3} T^{2} \)
29 \( 1 + 166 T + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 + 450 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 - 180 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + 590 T + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + p^{3} T^{2} \)
67 \( 1 - 740 T + p^{3} T^{2} \)
71 \( 1 - 688 T + p^{3} T^{2} \)
73 \( 1 + p^{3} T^{2} \)
79 \( 1 + 1384 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + p^{3} T^{2} \)
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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

−8.784853078304363255888346571584, −7.85042915357328648786477403234, −7.08012648942760879408187945070, −6.03074505577547228334097971392, −5.35839426906832944408285085560, −4.87919502475217459673920068436, −3.60814427670431172414485522132, −2.88453729935879048586939345379, −2.06407466010492228484826349351, 0, 2.06407466010492228484826349351, 2.88453729935879048586939345379, 3.60814427670431172414485522132, 4.87919502475217459673920068436, 5.35839426906832944408285085560, 6.03074505577547228334097971392, 7.08012648942760879408187945070, 7.85042915357328648786477403234, 8.784853078304363255888346571584

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