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Properties

Label 2-124950-1.1-c1-0-235
Degree $2$
Conductor $124950$
Sign $1$
Analytic cond. $997.730$
Root an. cond. $31.5868$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4·11-s + 12-s − 2·13-s + 16-s + 17-s − 18-s − 4·19-s + 4·22-s − 24-s + 2·26-s + 27-s − 10·29-s − 8·31-s − 32-s − 4·33-s − 34-s + 36-s + 2·37-s + 4·38-s − 2·39-s − 10·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.852·22-s − 0.204·24-s + 0.392·26-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.696·33-s − 0.171·34-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.320·39-s − 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(997.730\)
Root analytic conductor: \(31.5868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 124950,\ (\ :1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.06859735102406, −13.33856739120204, −13.07693518076529, −12.65359264033312, −12.11029286594421, −11.46041005214309, −10.93962152804514, −10.62977185821207, −9.999835346166744, −9.600892091860405, −9.233113922909348, −8.480606474444227, −8.192055481722036, −7.736033964956661, −7.187002255619248, −6.793609858685189, −6.116052370267694, −5.315040120933438, −5.150905921661023, −4.302457088115447, −3.487841463885431, −3.257283452304588, −2.243382835778207, −2.095309885092923, −1.344544746055745, 0, 0, 1.344544746055745, 2.095309885092923, 2.243382835778207, 3.257283452304588, 3.487841463885431, 4.302457088115447, 5.150905921661023, 5.315040120933438, 6.116052370267694, 6.793609858685189, 7.187002255619248, 7.736033964956661, 8.192055481722036, 8.480606474444227, 9.233113922909348, 9.600892091860405, 9.999835346166744, 10.62977185821207, 10.93962152804514, 11.46041005214309, 12.11029286594421, 12.65359264033312, 13.07693518076529, 13.33856739120204, 14.06859735102406

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