Properties

Label 2-950-1.1-c1-0-24
Degree $2$
Conductor $950$
Sign $1$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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 + 2.56·3-s + 4-s + 2.56·6-s + 2.56·7-s + 8-s + 3.56·9-s + 4·11-s + 2.56·12-s − 5.68·13-s + 2.56·14-s + 16-s − 3.43·17-s + 3.56·18-s − 19-s + 6.56·21-s + 4·22-s − 7.68·23-s + 2.56·24-s − 5.68·26-s + 1.43·27-s + 2.56·28-s − 5.68·29-s − 5.12·31-s + 32-s + 10.2·33-s − 3.43·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.47·3-s + 0.5·4-s + 1.04·6-s + 0.968·7-s + 0.353·8-s + 1.18·9-s + 1.20·11-s + 0.739·12-s − 1.57·13-s + 0.684·14-s + 0.250·16-s − 0.833·17-s + 0.839·18-s − 0.229·19-s + 1.43·21-s + 0.852·22-s − 1.60·23-s + 0.522·24-s − 1.11·26-s + 0.276·27-s + 0.484·28-s − 1.05·29-s − 0.920·31-s + 0.176·32-s + 1.78·33-s − 0.589·34-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: no
Arithmetic: yes
Character: $\chi_{950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.126092246\)
\(L(\frac12)\) \(\approx\) \(4.126092246\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 3T^{2} \)
7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 5.68T + 13T^{2} \)
17 \( 1 + 3.43T + 17T^{2} \)
23 \( 1 + 7.68T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 - 2.87T + 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 - 2.56T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 2.56T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 1.68T + 73T^{2} \)
79 \( 1 + 5.12T + 79T^{2} \)
83 \( 1 + 2.87T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.730747983450564972363666958319, −9.269461802051136599333512278135, −8.218481723170876935839658217475, −7.63377684795684572540054953690, −6.80217316125886040642651172131, −5.56507900355847824074066298697, −4.34525009015419484453738364839, −3.90135760216719732222356982045, −2.48187998105717011530260448552, −1.88554258137734670923623349210, 1.88554258137734670923623349210, 2.48187998105717011530260448552, 3.90135760216719732222356982045, 4.34525009015419484453738364839, 5.56507900355847824074066298697, 6.80217316125886040642651172131, 7.63377684795684572540054953690, 8.218481723170876935839658217475, 9.269461802051136599333512278135, 9.730747983450564972363666958319

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