Properties

Label 2-19950-1.1-c1-0-60
Degree $2$
Conductor $19950$
Sign $-1$
Analytic cond. $159.301$
Root an. cond. $12.6214$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 7-s − 8-s + 9-s − 2·11-s − 12-s + 5·13-s − 14-s + 16-s − 17-s − 18-s + 19-s − 21-s + 2·22-s + 5·23-s + 24-s − 5·26-s − 27-s + 28-s + 9·29-s + 3·31-s − 32-s + 2·33-s + 34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.218·21-s + 0.426·22-s + 1.04·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.538·31-s − 0.176·32-s + 0.348·33-s + 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(159.301\)
Root analytic conductor: \(12.6214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 19950,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.78570938822357, −15.66120552581310, −15.16750467553315, −14.25806978351640, −13.68833498181171, −13.23623180043995, −12.56744808628541, −11.76689404237077, −11.63355503439997, −10.80529843035086, −10.37160569926445, −10.13713789935495, −8.974354376563488, −8.728634841564540, −8.182389143852626, −7.386025662492733, −6.878336146380022, −6.247222654638402, −5.650627965032380, −4.944124408019179, −4.343180711038741, −3.322880639069419, −2.760388542924201, −1.602235755037530, −1.109574347031903, 0, 1.109574347031903, 1.602235755037530, 2.760388542924201, 3.322880639069419, 4.343180711038741, 4.944124408019179, 5.650627965032380, 6.247222654638402, 6.878336146380022, 7.386025662492733, 8.182389143852626, 8.728634841564540, 8.974354376563488, 10.13713789935495, 10.37160569926445, 10.80529843035086, 11.63355503439997, 11.76689404237077, 12.56744808628541, 13.23623180043995, 13.68833498181171, 14.25806978351640, 15.16750467553315, 15.66120552581310, 15.78570938822357

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