Properties

Label 2-750-1.1-c1-0-11
Degree $2$
Conductor $750$
Sign $-1$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 1.61·7-s − 8-s + 9-s + 0.618·11-s − 12-s + 1.85·13-s + 1.61·14-s + 16-s − 5.23·17-s − 18-s − 0.854·19-s + 1.61·21-s − 0.618·22-s + 1.85·23-s + 24-s − 1.85·26-s − 27-s − 1.61·28-s − 7.23·29-s + 6.47·31-s − 32-s − 0.618·33-s + 5.23·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.611·7-s − 0.353·8-s + 0.333·9-s + 0.186·11-s − 0.288·12-s + 0.514·13-s + 0.432·14-s + 0.250·16-s − 1.26·17-s − 0.235·18-s − 0.195·19-s + 0.353·21-s − 0.131·22-s + 0.386·23-s + 0.204·24-s − 0.363·26-s − 0.192·27-s − 0.305·28-s − 1.34·29-s + 1.16·31-s − 0.176·32-s − 0.107·33-s + 0.897·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 - 0.618T + 11T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 + 0.854T + 19T^{2} \)
23 \( 1 - 1.85T + 23T^{2} \)
29 \( 1 + 7.23T + 29T^{2} \)
31 \( 1 - 6.47T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 - 7.70T + 43T^{2} \)
47 \( 1 - 0.618T + 47T^{2} \)
53 \( 1 + 7.61T + 53T^{2} \)
59 \( 1 + 1.90T + 59T^{2} \)
61 \( 1 - 3.70T + 61T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 4.94T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 2.94T + 83T^{2} \)
89 \( 1 + 6.90T + 89T^{2} \)
97 \( 1 - 3.70T + 97T^{2} \)
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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

−9.951473967070916097461052877444, −9.086836040638722481835154795515, −8.398030197634529302795165248625, −7.15073673323062541880957356011, −6.56743490124000774670703675385, −5.68736614296276151981811890139, −4.43505676390634117498281156287, −3.19552408320630230696342379817, −1.70611611768860655788354343983, 0, 1.70611611768860655788354343983, 3.19552408320630230696342379817, 4.43505676390634117498281156287, 5.68736614296276151981811890139, 6.56743490124000774670703675385, 7.15073673323062541880957356011, 8.398030197634529302795165248625, 9.086836040638722481835154795515, 9.951473967070916097461052877444

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