Properties

Label 2-950-1.1-c1-0-22
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 + 3.25·3-s + 4-s + 3.25·6-s − 0.0778·7-s + 8-s + 7.58·9-s − 4.50·11-s + 3.25·12-s − 5.33·13-s − 0.0778·14-s + 16-s + 7.33·17-s + 7.58·18-s + 19-s − 0.253·21-s − 4.50·22-s + 3.40·23-s + 3.25·24-s − 5.33·26-s + 14.9·27-s − 0.0778·28-s − 1.33·29-s − 2.50·31-s + 32-s − 14.6·33-s + 7.33·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.87·3-s + 0.5·4-s + 1.32·6-s − 0.0294·7-s + 0.353·8-s + 2.52·9-s − 1.35·11-s + 0.939·12-s − 1.47·13-s − 0.0208·14-s + 0.250·16-s + 1.77·17-s + 1.78·18-s + 0.229·19-s − 0.0553·21-s − 0.960·22-s + 0.710·23-s + 0.664·24-s − 1.04·26-s + 2.87·27-s − 0.0147·28-s − 0.247·29-s − 0.450·31-s + 0.176·32-s − 2.55·33-s + 1.25·34-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

\(L(1)\) \(\approx\) \(4.283853217\)
\(L(\frac12)\) \(\approx\) \(4.283853217\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.25T + 3T^{2} \)
7 \( 1 + 0.0778T + 7T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 + 5.33T + 13T^{2} \)
17 \( 1 - 7.33T + 17T^{2} \)
23 \( 1 - 3.40T + 23T^{2} \)
29 \( 1 + 1.33T + 29T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 0.506T + 43T^{2} \)
47 \( 1 + 5.66T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 7.56T + 59T^{2} \)
61 \( 1 + 2.15T + 61T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 5.09T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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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

−9.932423726613245696562613802216, −9.284394690746707841738460036270, −8.057421319368204439048348922864, −7.69227950036236084393324339981, −6.98333121191809933373799131855, −5.38394459315033954036635176639, −4.66249845069668929723597799206, −3.31677396625629183760648040160, −2.91352497355473632961713277846, −1.82307224634606915650469816414, 1.82307224634606915650469816414, 2.91352497355473632961713277846, 3.31677396625629183760648040160, 4.66249845069668929723597799206, 5.38394459315033954036635176639, 6.98333121191809933373799131855, 7.69227950036236084393324339981, 8.057421319368204439048348922864, 9.284394690746707841738460036270, 9.932423726613245696562613802216

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