Properties

Label 2-950-1.1-c1-0-13
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.76·3-s + 4-s + 2.76·6-s + 0.761·7-s − 8-s + 4.62·9-s − 0.864·11-s − 2.76·12-s − 5.62·13-s − 0.761·14-s + 16-s + 3.62·17-s − 4.62·18-s + 19-s − 2.10·21-s + 0.864·22-s + 8.01·23-s + 2.76·24-s + 5.62·26-s − 4.49·27-s + 0.761·28-s − 7.35·29-s + 8.11·31-s − 32-s + 2.38·33-s − 3.62·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.59·3-s + 0.5·4-s + 1.12·6-s + 0.287·7-s − 0.353·8-s + 1.54·9-s − 0.260·11-s − 0.797·12-s − 1.56·13-s − 0.203·14-s + 0.250·16-s + 0.879·17-s − 1.09·18-s + 0.229·19-s − 0.458·21-s + 0.184·22-s + 1.67·23-s + 0.563·24-s + 1.10·26-s − 0.864·27-s + 0.143·28-s − 1.36·29-s + 1.45·31-s − 0.176·32-s + 0.415·33-s − 0.621·34-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: \(1\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 2.76T + 3T^{2} \)
7 \( 1 - 0.761T + 7T^{2} \)
11 \( 1 + 0.864T + 11T^{2} \)
13 \( 1 + 5.62T + 13T^{2} \)
17 \( 1 - 3.62T + 17T^{2} \)
23 \( 1 - 8.01T + 23T^{2} \)
29 \( 1 + 7.35T + 29T^{2} \)
31 \( 1 - 8.11T + 31T^{2} \)
37 \( 1 + 0.476T + 37T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 + 6.86T + 43T^{2} \)
47 \( 1 + 1.25T + 47T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 - 4.49T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 1.03T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 0.270T + 83T^{2} \)
89 \( 1 - 0.387T + 89T^{2} \)
97 \( 1 - 8.50T + 97T^{2} \)
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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

−9.984039453698283500953277843232, −8.928598727444693045189016575900, −7.67320077953484857098807346540, −7.16067638001083096200682895369, −6.21227652886072284894188099584, −5.24993733857020100458888328404, −4.73324696949202920321370441690, −2.96927646352108528622151749854, −1.37919515776564355708143992288, 0, 1.37919515776564355708143992288, 2.96927646352108528622151749854, 4.73324696949202920321370441690, 5.24993733857020100458888328404, 6.21227652886072284894188099584, 7.16067638001083096200682895369, 7.67320077953484857098807346540, 8.928598727444693045189016575900, 9.984039453698283500953277843232

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