Properties

Label 2-950-1.1-c1-0-3
Degree $2$
Conductor $950$
Sign $1$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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 + 7-s − 8-s − 2·9-s − 12-s + 13-s − 14-s + 16-s + 3·17-s + 2·18-s + 19-s − 21-s − 3·23-s + 24-s − 26-s + 5·27-s + 28-s − 3·29-s + 2·31-s − 32-s − 3·34-s − 2·36-s + 10·37-s − 38-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 − 2/3·9-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.218·21-s − 0.625·23-s + 0.204·24-s − 0.196·26-s + 0.962·27-s + 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 0.514·34-s − 1/3·36-s + 1.64·37-s − 0.162·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8821121233\)
\(L(\frac12)\) \(\approx\) \(0.8821121233\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.983035167615231513039157588703, −9.303912250997036448752264323167, −8.237637220939037195909198183700, −7.77982432388455298937264149525, −6.57249513174536622080526940747, −5.85301595783600076781492520747, −4.98057051522020806813162729411, −3.61241990046837984333714647492, −2.34024048259330359076586962577, −0.850809003224210732230405050623, 0.850809003224210732230405050623, 2.34024048259330359076586962577, 3.61241990046837984333714647492, 4.98057051522020806813162729411, 5.85301595783600076781492520747, 6.57249513174536622080526940747, 7.77982432388455298937264149525, 8.237637220939037195909198183700, 9.303912250997036448752264323167, 9.983035167615231513039157588703

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