Properties

Label 2-124950-1.1-c1-0-83
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 2·11-s − 12-s − 2·13-s + 16-s + 17-s + 18-s + 4·19-s + 2·22-s − 2·23-s − 24-s − 2·26-s − 27-s + 2·29-s + 8·31-s + 32-s − 2·33-s + 34-s + 36-s − 6·37-s + 4·38-s + 2·39-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 + 0.603·11-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 + 0.426·22-s − 0.417·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s − 0.986·37-s + 0.648·38-s + 0.320·39-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: \(0\)
Selberg data: \((2,\ 124950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.138644604\)
\(L(\frac12)\) \(\approx\) \(4.138644604\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.76288559589054, −12.82410486007503, −12.58381634490625, −12.10280013946011, −11.66148162112604, −11.30767746648522, −10.74015872052921, −10.15982678521817, −9.698797553070950, −9.365884789864169, −8.505427440789754, −8.011970668962756, −7.498968472447986, −6.840855295169455, −6.584635825795241, −5.933487837025105, −5.430960567277713, −4.964051139195012, −4.423188028541240, −3.867161585895447, −3.304403562257252, −2.609084481178805, −2.020333934670294, −1.161015918093881, −0.6272328222937177, 0.6272328222937177, 1.161015918093881, 2.020333934670294, 2.609084481178805, 3.304403562257252, 3.867161585895447, 4.423188028541240, 4.964051139195012, 5.430960567277713, 5.933487837025105, 6.584635825795241, 6.840855295169455, 7.498968472447986, 8.011970668962756, 8.505427440789754, 9.365884789864169, 9.698797553070950, 10.15982678521817, 10.74015872052921, 11.30767746648522, 11.66148162112604, 12.10280013946011, 12.58381634490625, 12.82410486007503, 13.76288559589054

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