Properties

Label 2-1950-1.1-c1-0-4
Degree $2$
Conductor $1950$
Sign $1$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 − 4·7-s − 8-s + 9-s + 12-s + 13-s + 4·14-s + 16-s − 18-s + 5·19-s − 4·21-s − 24-s − 26-s + 27-s − 4·28-s + 3·29-s − 4·31-s − 32-s + 36-s − 7·37-s − 5·38-s + 39-s + 3·41-s + 4·42-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s + 1.14·19-s − 0.872·21-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.755·28-s + 0.557·29-s − 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.15·37-s − 0.811·38-s + 0.160·39-s + 0.468·41-s + 0.617·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.261653142\)
\(L(\frac12)\) \(\approx\) \(1.261653142\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.078339016738156498544771857711, −8.696657713782855691873839819406, −7.55763254790499154685263093337, −7.04556559344328990168385895474, −6.21942354210226036235913961670, −5.34655100340507626856083927354, −3.86541777052856340745686880083, −3.21110717019141316546603900405, −2.28203067662838759760760090270, −0.805107178230786108851722746580, 0.805107178230786108851722746580, 2.28203067662838759760760090270, 3.21110717019141316546603900405, 3.86541777052856340745686880083, 5.34655100340507626856083927354, 6.21942354210226036235913961670, 7.04556559344328990168385895474, 7.55763254790499154685263093337, 8.696657713782855691873839819406, 9.078339016738156498544771857711

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