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Properties

Label 2-475-19.11-c1-0-20
Degree $2$
Conductor $475$
Sign $0.0977 + 0.995i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−0.999 − 1.73i)4-s + 4·7-s + (1.5 + 2.59i)9-s − 11-s + (−1 − 1.73i)13-s + (4 − 6.92i)14-s + (1.99 − 3.46i)16-s + (−1 + 1.73i)17-s + 6·18-s + (−3.5 − 2.59i)19-s + (−1 + 1.73i)22-s + (3 + 5.19i)23-s − 3.99·26-s + (−3.99 − 6.92i)28-s + (−4.5 − 7.79i)29-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + 1.51·7-s + (0.5 + 0.866i)9-s − 0.301·11-s + (−0.277 − 0.480i)13-s + (1.06 − 1.85i)14-s + (0.499 − 0.866i)16-s + (−0.242 + 0.420i)17-s + 1.41·18-s + (−0.802 − 0.596i)19-s + (−0.213 + 0.369i)22-s + (0.625 + 1.08i)23-s − 0.784·26-s + (−0.755 − 1.30i)28-s + (−0.835 − 1.44i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0977 + 0.995i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75517 - 1.59125i\)
\(L(\frac12)\) \(\approx\) \(1.75517 - 1.59125i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-5.5 - 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3 - 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.90471578686655770881722212476, −10.47932195585320547667850945242, −9.231403414712642924893437783538, −7.935791464766070634966570650783, −7.43139367874779369906396395452, −5.53545734999154402714993023198, −4.80195433734292026316143436394, −3.98980113475805215523919362745, −2.45324272264237617901690965065, −1.60205579459502770893204042843, 1.77629153868512302240926579949, 3.85089679886022452512250861402, 4.75128138598538874450553548668, 5.47336379223912505359138887712, 6.70759651690475235308707442941, 7.28682255474782080712788413973, 8.284883554916736610418978676258, 9.066647111327061047407457202854, 10.47926428389554250498594186373, 11.21206814392667914991093789148

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