Properties

Label 2-475-1.1-c1-0-5
Degree $2$
Conductor $475$
Sign $1$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  − 1.95·2-s − 0.296·3-s + 1.82·4-s + 0.580·6-s + 3.56·7-s + 0.340·8-s − 2.91·9-s + 5.56·11-s − 0.541·12-s − 5.26·13-s − 6.96·14-s − 4.31·16-s − 1.40·17-s + 5.69·18-s + 19-s − 1.05·21-s − 10.8·22-s + 6.96·23-s − 0.101·24-s + 10.3·26-s + 1.75·27-s + 6.50·28-s + 1.40·29-s + 1.75·31-s + 7.76·32-s − 1.65·33-s + 2.75·34-s + ⋯
L(s)  = 1  − 1.38·2-s − 0.171·3-s + 0.912·4-s + 0.237·6-s + 1.34·7-s + 0.120·8-s − 0.970·9-s + 1.67·11-s − 0.156·12-s − 1.46·13-s − 1.86·14-s − 1.07·16-s − 0.341·17-s + 1.34·18-s + 0.229·19-s − 0.230·21-s − 2.31·22-s + 1.45·23-s − 0.0206·24-s + 2.02·26-s + 0.337·27-s + 1.22·28-s + 0.261·29-s + 0.315·31-s + 1.37·32-s − 0.287·33-s + 0.471·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7346313899\)
\(L(\frac12)\) \(\approx\) \(0.7346313899\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 - T \)
good2 \( 1 + 1.95T + 2T^{2} \)
3 \( 1 + 0.296T + 3T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
11 \( 1 - 5.56T + 11T^{2} \)
13 \( 1 + 5.26T + 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
23 \( 1 - 6.96T + 23T^{2} \)
29 \( 1 - 1.40T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + 3.61T + 37T^{2} \)
41 \( 1 - 4.34T + 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 - 8.26T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 - 9.47T + 59T^{2} \)
61 \( 1 - 9.21T + 61T^{2} \)
67 \( 1 - 4.76T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + 6.59T + 73T^{2} \)
79 \( 1 - 5.47T + 79T^{2} \)
83 \( 1 - 4.15T + 83T^{2} \)
89 \( 1 + 9.23T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−11.03440528866184541543097749095, −9.970257018727239853744647533509, −8.986479645325782915771213959967, −8.623357474022990250994013914137, −7.50887323311184825290869264057, −6.81889230570599300259343450750, −5.33379504728974060162794953732, −4.34645696729541782834819138731, −2.40159277881274627052346039447, −1.03888422211361464049850295982, 1.03888422211361464049850295982, 2.40159277881274627052346039447, 4.34645696729541782834819138731, 5.33379504728974060162794953732, 6.81889230570599300259343450750, 7.50887323311184825290869264057, 8.623357474022990250994013914137, 8.986479645325782915771213959967, 9.970257018727239853744647533509, 11.03440528866184541543097749095

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