Properties

Label 2-35e2-1.1-c3-0-74
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-s − 6·3-s − 7·4-s + 6·6-s + 15·8-s + 9·9-s − 44·11-s + 42·12-s − 6·13-s + 41·16-s + 24·17-s − 9·18-s − 114·19-s + 44·22-s + 52·23-s − 90·24-s + 6·26-s + 108·27-s + 146·29-s − 276·31-s − 161·32-s + 264·33-s − 24·34-s − 63·36-s + 210·37-s + 114·38-s + 36·39-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.15·3-s − 7/8·4-s + 0.408·6-s + 0.662·8-s + 1/3·9-s − 1.20·11-s + 1.01·12-s − 0.128·13-s + 0.640·16-s + 0.342·17-s − 0.117·18-s − 1.37·19-s + 0.426·22-s + 0.471·23-s − 0.765·24-s + 0.0452·26-s + 0.769·27-s + 0.934·29-s − 1.59·31-s − 0.889·32-s + 1.39·33-s − 0.121·34-s − 0.291·36-s + 0.933·37-s + 0.486·38-s + 0.147·39-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: Trivial
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 \( 1 + T + p^{3} T^{2} \)
3 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 + 4 p T + p^{3} T^{2} \)
13 \( 1 + 6 T + p^{3} T^{2} \)
17 \( 1 - 24 T + p^{3} T^{2} \)
19 \( 1 + 6 p T + p^{3} T^{2} \)
23 \( 1 - 52 T + p^{3} T^{2} \)
29 \( 1 - 146 T + p^{3} T^{2} \)
31 \( 1 + 276 T + p^{3} T^{2} \)
37 \( 1 - 210 T + p^{3} T^{2} \)
41 \( 1 - 444 T + p^{3} T^{2} \)
43 \( 1 + 492 T + p^{3} T^{2} \)
47 \( 1 - 612 T + p^{3} T^{2} \)
53 \( 1 + 50 T + p^{3} T^{2} \)
59 \( 1 - 294 T + p^{3} T^{2} \)
61 \( 1 - 450 T + p^{3} T^{2} \)
67 \( 1 - 668 T + p^{3} T^{2} \)
71 \( 1 + 308 T + p^{3} T^{2} \)
73 \( 1 + 12 T + p^{3} T^{2} \)
79 \( 1 - 596 T + p^{3} T^{2} \)
83 \( 1 - 966 T + p^{3} T^{2} \)
89 \( 1 + 408 T + p^{3} T^{2} \)
97 \( 1 - 1200 T + p^{3} T^{2} \)
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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

−8.929937775715720046282148906364, −8.193925112538205252526934607248, −7.36871676476285734508160469467, −6.29304914570146111363098699606, −5.42125784287822869450423007263, −4.89410289698218912035753941136, −3.90703510822983577721829068950, −2.45184323825755262856874000499, −0.858508408614729945764807493855, 0, 0.858508408614729945764807493855, 2.45184323825755262856874000499, 3.90703510822983577721829068950, 4.89410289698218912035753941136, 5.42125784287822869450423007263, 6.29304914570146111363098699606, 7.36871676476285734508160469467, 8.193925112538205252526934607248, 8.929937775715720046282148906364

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