Properties

Label 2-1950-1.1-c1-0-2
Degree $2$
Conductor $1950$
Sign $1$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational no
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 − 4.70·7-s − 8-s + 9-s + 4.70·11-s − 12-s + 13-s + 4.70·14-s + 16-s + 0.701·17-s − 18-s − 1.70·19-s + 4.70·21-s − 4.70·22-s + 24-s − 26-s − 27-s − 4.70·28-s − 6.40·29-s − 10.1·31-s − 32-s − 4.70·33-s − 0.701·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.77·7-s − 0.353·8-s + 0.333·9-s + 1.41·11-s − 0.288·12-s + 0.277·13-s + 1.25·14-s + 0.250·16-s + 0.170·17-s − 0.235·18-s − 0.390·19-s + 1.02·21-s − 1.00·22-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.888·28-s − 1.18·29-s − 1.81·31-s − 0.176·32-s − 0.818·33-s − 0.120·34-s + 0.166·36-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: no
Arithmetic: yes
Character: $\chi_{1950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7018972008\)
\(L(\frac12)\) \(\approx\) \(0.7018972008\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.70T + 7T^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
17 \( 1 - 0.701T + 17T^{2} \)
19 \( 1 + 1.70T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6.40T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 1.70T + 37T^{2} \)
41 \( 1 + 3.70T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 2.40T + 53T^{2} \)
59 \( 1 - 2.70T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 6.40T + 67T^{2} \)
71 \( 1 + 1.70T + 71T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 + 5.70T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 2.59T + 97T^{2} \)
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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

−9.472787860436681976465719940768, −8.653466377789352576075070335817, −7.43037237241838655659166488594, −6.76591564220039910140805632749, −6.23684722455429737381600457336, −5.50645854659061271551715813891, −3.95070265072329045531588778973, −3.43051010611221387236148923197, −1.99845233811937189852535600610, −0.63040485423437105060734933614, 0.63040485423437105060734933614, 1.99845233811937189852535600610, 3.43051010611221387236148923197, 3.95070265072329045531588778973, 5.50645854659061271551715813891, 6.23684722455429737381600457336, 6.76591564220039910140805632749, 7.43037237241838655659166488594, 8.653466377789352576075070335817, 9.472787860436681976465719940768

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