Properties

Label 2-1950-1.1-c1-0-35
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 1.23·7-s + 8-s + 9-s − 4.47·11-s − 12-s − 13-s + 1.23·14-s + 16-s − 2.76·17-s + 18-s − 7.23·19-s − 1.23·21-s − 4.47·22-s − 7.23·23-s − 24-s − 26-s − 27-s + 1.23·28-s + 9.70·29-s − 4·31-s + 32-s + 4.47·33-s − 2.76·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.467·7-s + 0.353·8-s + 0.333·9-s − 1.34·11-s − 0.288·12-s − 0.277·13-s + 0.330·14-s + 0.250·16-s − 0.670·17-s + 0.235·18-s − 1.66·19-s − 0.269·21-s − 0.953·22-s − 1.50·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.233·28-s + 1.80·29-s − 0.718·31-s + 0.176·32-s + 0.778·33-s − 0.474·34-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: \(1\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
17 \( 1 + 2.76T + 17T^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 - 9.70T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 6.94T + 37T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + 6.47T + 43T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 - 0.472T + 59T^{2} \)
61 \( 1 - 6.94T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 + 8.76T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + 9.70T + 97T^{2} \)
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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

−8.530779694568573213995301552174, −8.004332164035201635894589716494, −7.03845459460414655963082095928, −6.26948536369219982766681897972, −5.51991667166007308496010389151, −4.69024563761773266461927590004, −4.12832467108517016568004614766, −2.71788715017534627395376652419, −1.88660992270747490487052641091, 0, 1.88660992270747490487052641091, 2.71788715017534627395376652419, 4.12832467108517016568004614766, 4.69024563761773266461927590004, 5.51991667166007308496010389151, 6.26948536369219982766681897972, 7.03845459460414655963082095928, 8.004332164035201635894589716494, 8.530779694568573213995301552174

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