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Properties

Label 2-5175-1.1-c1-0-157
Degree $2$
Conductor $5175$
Sign $-1$
Analytic cond. $41.3225$
Root an. cond. $6.42826$
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  + 2·2-s + 2·4-s − 7-s − 2·13-s − 2·14-s − 4·16-s − 5·17-s + 8·19-s − 23-s − 4·26-s − 2·28-s + 5·29-s − 5·31-s − 8·32-s − 10·34-s − 7·37-s + 16·38-s + 7·41-s − 4·43-s − 2·46-s − 2·47-s − 6·49-s − 4·52-s − 53-s + 10·58-s − 3·59-s − 6·61-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.377·7-s − 0.554·13-s − 0.534·14-s − 16-s − 1.21·17-s + 1.83·19-s − 0.208·23-s − 0.784·26-s − 0.377·28-s + 0.928·29-s − 0.898·31-s − 1.41·32-s − 1.71·34-s − 1.15·37-s + 2.59·38-s + 1.09·41-s − 0.609·43-s − 0.294·46-s − 0.291·47-s − 6/7·49-s − 0.554·52-s − 0.137·53-s + 1.31·58-s − 0.390·59-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5175\)    =    \(3^{2} \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(41.3225\)
Root analytic conductor: \(6.42826\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5175,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.48507948380278768811282018027, −7.01288317299784914067740362367, −6.19726166274718348936741924112, −5.58050424077601487859746395299, −4.81579420738327512528197627273, −4.29320776196952273000482135787, −3.26125120401248575653520604420, −2.84739911677715532341125970534, −1.69177605583615931536111172975, 0, 1.69177605583615931536111172975, 2.84739911677715532341125970534, 3.26125120401248575653520604420, 4.29320776196952273000482135787, 4.81579420738327512528197627273, 5.58050424077601487859746395299, 6.19726166274718348936741924112, 7.01288317299784914067740362367, 7.48507948380278768811282018027

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