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Properties

Label 2-475-19.18-c0-0-2
Degree $2$
Conductor $475$
Sign $-1$
Analytic cond. $0.237055$
Root an. cond. $0.486883$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 2.00·6-s − 1.00·9-s + 1.41i·12-s + 1.41i·13-s − 0.999·16-s + 1.41i·18-s + 19-s + 2.00·26-s + 1.41i·32-s + 1.00·36-s + 1.41i·37-s − 1.41i·38-s + 2.00·39-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 2.00·6-s − 1.00·9-s + 1.41i·12-s + 1.41i·13-s − 0.999·16-s + 1.41i·18-s + 19-s + 2.00·26-s + 1.41i·32-s + 1.00·36-s + 1.41i·37-s − 1.41i·38-s + 2.00·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(0.237055\)
Root analytic conductor: \(0.486883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8608239639\)
\(L(\frac12)\) \(\approx\) \(0.8608239639\)
\(L(1)\) 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 \( 1 + 1.41iT - T^{2} \)
3 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−11.17766425103694944005914721705, −9.975729415498283986647053493887, −9.217191113077171857790777816209, −8.111811888096179358965101859191, −7.03731933014322027177403202566, −6.38299027142892247989561511415, −4.77822955115321017582691481289, −3.44325559732078420873714628887, −2.22458540386151061200000898513, −1.31743840664703523636206152464, 3.05790241124699620181144792259, 4.29637028922848141649342123691, 5.30219161385696357359246288479, 5.81023703002536076086294186153, 7.17561271918665208873725262681, 7.994480307557649378855852847443, 8.920466863220392946068575919118, 9.718099286941911623165117393099, 10.57623719591102067851626240796, 11.37990582308993058538540547590

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