Properties

Label 2-1950-1.1-c1-0-21
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 + 3·7-s + 8-s + 9-s − 3·11-s + 12-s + 13-s + 3·14-s + 16-s + 3·17-s + 18-s + 3·21-s − 3·22-s + 4·23-s + 24-s + 26-s + 27-s + 3·28-s + 5·29-s − 3·31-s + 32-s − 3·33-s + 3·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.654·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.566·28-s + 0.928·29-s − 0.538·31-s + 0.176·32-s − 0.522·33-s + 0.514·34-s + 1/6·36-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: yes
Arithmetic: yes
Character: $\chi_{1950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.885621465\)
\(L(\frac12)\) \(\approx\) \(3.885621465\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.933816175732799236793232920481, −8.372178951472568303607979391931, −7.55697011933255808289536752220, −7.00227519035662633790722146486, −5.70721380824909254302815531777, −5.12549501155078775052077320989, −4.29885874464191948579845964703, −3.29845174761783363418427055475, −2.41024263072843413513774850848, −1.32512616341537127516841939290, 1.32512616341537127516841939290, 2.41024263072843413513774850848, 3.29845174761783363418427055475, 4.29885874464191948579845964703, 5.12549501155078775052077320989, 5.70721380824909254302815531777, 7.00227519035662633790722146486, 7.55697011933255808289536752220, 8.372178951472568303607979391931, 8.933816175732799236793232920481

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