Properties

Label 2-950-1.1-c1-0-17
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 − 1.36·3-s + 4-s + 1.36·6-s − 0.636·7-s − 8-s − 1.14·9-s + 3.50·11-s − 1.36·12-s + 0.141·13-s + 0.636·14-s + 16-s − 2.14·17-s + 1.14·18-s + 19-s + 0.867·21-s − 3.50·22-s − 4.91·23-s + 1.36·24-s − 0.141·26-s + 5.64·27-s − 0.636·28-s + 7.15·29-s − 7.78·31-s − 32-s − 4.77·33-s + 2.14·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.787·3-s + 0.5·4-s + 0.556·6-s − 0.240·7-s − 0.353·8-s − 0.380·9-s + 1.05·11-s − 0.393·12-s + 0.0391·13-s + 0.170·14-s + 0.250·16-s − 0.519·17-s + 0.269·18-s + 0.229·19-s + 0.189·21-s − 0.747·22-s − 1.02·23-s + 0.278·24-s − 0.0277·26-s + 1.08·27-s − 0.120·28-s + 1.32·29-s − 1.39·31-s − 0.176·32-s − 0.831·33-s + 0.367·34-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 \( 1 + 1.36T + 3T^{2} \)
7 \( 1 + 0.636T + 7T^{2} \)
11 \( 1 - 3.50T + 11T^{2} \)
13 \( 1 - 0.141T + 13T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
23 \( 1 + 4.91T + 23T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 + 3.27T + 37T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 8.14T + 53T^{2} \)
59 \( 1 + 5.64T + 59T^{2} \)
61 \( 1 + 6.49T + 61T^{2} \)
67 \( 1 + 8.37T + 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 - 3.69T + 73T^{2} \)
79 \( 1 + 4.17T + 79T^{2} \)
83 \( 1 - 9.00T + 83T^{2} \)
89 \( 1 + 6.77T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−9.570129306125444829423161640153, −8.893607415941541781459788532898, −8.060076202098181060261049926615, −6.91417056560953214042847626485, −6.33009541371317790535834993373, −5.49942535587784043290781602513, −4.28342306821144394432530696198, −3.04008517123061672221722669610, −1.55564356243632533712808947973, 0, 1.55564356243632533712808947973, 3.04008517123061672221722669610, 4.28342306821144394432530696198, 5.49942535587784043290781602513, 6.33009541371317790535834993373, 6.91417056560953214042847626485, 8.060076202098181060261049926615, 8.893607415941541781459788532898, 9.570129306125444829423161640153

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