Properties

Label 2-1575-1.1-c1-0-31
Degree $2$
Conductor $1575$
Sign $-1$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
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 − 4-s + 7-s + 3·8-s − 4·11-s + 2·13-s − 14-s − 16-s − 6·17-s + 4·19-s + 4·22-s − 2·26-s − 28-s + 2·29-s − 5·32-s + 6·34-s − 6·37-s − 4·38-s − 2·41-s + 4·43-s + 4·44-s + 49-s − 2·52-s + 6·53-s + 3·56-s − 2·58-s − 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.20·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.852·22-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 1/7·49-s − 0.277·52-s + 0.824·53-s + 0.400·56-s − 0.262·58-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.941556964198447272384176621072, −8.352294990335304362757323779032, −7.65494363024181651773191318169, −6.85360812870284917864517668882, −5.61533530390354367379245545667, −4.87983646890892800864940889103, −4.03796996879678948025961788837, −2.72775334283054769167882483900, −1.46036150187712502072371434143, 0, 1.46036150187712502072371434143, 2.72775334283054769167882483900, 4.03796996879678948025961788837, 4.87983646890892800864940889103, 5.61533530390354367379245545667, 6.85360812870284917864517668882, 7.65494363024181651773191318169, 8.352294990335304362757323779032, 8.941556964198447272384176621072

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