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Properties

Label 2-475-95.94-c2-0-54
Degree $2$
Conductor $475$
Sign $0.989 + 0.141i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.60·2-s + 3.60·3-s + 8.99·4-s + 12.9·6-s − 5i·7-s + 18.0·8-s + 3.99·9-s − 10·11-s + 32.4·12-s − 3.60·13-s − 18.0i·14-s + 28.9·16-s + 15i·17-s + 14.4·18-s + (6 + 18.0i)19-s + ⋯
L(s)  = 1  + 1.80·2-s + 1.20·3-s + 2.24·4-s + 2.16·6-s − 0.714i·7-s + 2.25·8-s + 0.444·9-s − 0.909·11-s + 2.70·12-s − 0.277·13-s − 1.28i·14-s + 1.81·16-s + 0.882i·17-s + 0.801·18-s + (0.315 + 0.948i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.989 + 0.141i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ 0.989 + 0.141i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.803261843\)
\(L(\frac12)\) \(\approx\) \(6.803261843\)
\(L(2)\) 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 \( 1 + (-6 - 18.0i)T \)
good2 \( 1 - 3.60T + 4T^{2} \)
3 \( 1 - 3.60T + 9T^{2} \)
7 \( 1 + 5iT - 49T^{2} \)
11 \( 1 + 10T + 121T^{2} \)
13 \( 1 + 3.60T + 169T^{2} \)
17 \( 1 - 15iT - 289T^{2} \)
23 \( 1 + 35iT - 529T^{2} \)
29 \( 1 - 18.0iT - 841T^{2} \)
31 \( 1 - 36.0iT - 961T^{2} \)
37 \( 1 + 21.6T + 1.36e3T^{2} \)
41 \( 1 + 36.0iT - 1.68e3T^{2} \)
43 \( 1 - 20iT - 1.84e3T^{2} \)
47 \( 1 - 10iT - 2.20e3T^{2} \)
53 \( 1 - 75.7T + 2.80e3T^{2} \)
59 \( 1 - 18.0iT - 3.48e3T^{2} \)
61 \( 1 + 40T + 3.72e3T^{2} \)
67 \( 1 - 39.6T + 4.48e3T^{2} \)
71 \( 1 + 108. iT - 5.04e3T^{2} \)
73 \( 1 + 105iT - 5.32e3T^{2} \)
79 \( 1 + 36.0iT - 6.24e3T^{2} \)
83 \( 1 - 40iT - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 122.T + 9.40e3T^{2} \)
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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

−10.73780773828025375459432670989, −10.32051056172351450305652948559, −8.750868568171052944542116320318, −7.81900525641326409971348173101, −7.00459733056120416094037082151, −5.87789743029339185414523297867, −4.80724642784834275126783989004, −3.81087305377229716940895857559, −3.04738396981920989676950008915, −1.99153945819978947106213986472, 2.30818173398764813827976122636, 2.80899820967120566905373291070, 3.84963676997206974078602232385, 5.08907327622472302920306048483, 5.72166928778086509262633407036, 7.10306209226862497409887226946, 7.80822325794979319846646500452, 8.989246667631569196899536875076, 9.926067835128235383602246565487, 11.36391024830292824329759193640

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