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Properties

Label 2-475-95.74-c1-0-26
Degree $2$
Conductor $475$
Sign $0.262 + 0.964i$
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  + (1.32 + 0.233i)2-s + (1.85 − 2.20i)3-s + (−0.173 − 0.0632i)4-s + (2.97 − 2.49i)6-s + (−0.300 + 0.173i)7-s + (−2.54 − 1.47i)8-s + (−0.918 − 5.21i)9-s + (1.11 − 1.92i)11-s + (−0.460 + 0.266i)12-s + (1.65 + 1.97i)13-s + (−0.439 + 0.160i)14-s + (−2.75 − 2.31i)16-s + (−0.460 − 0.0812i)17-s − 7.12i·18-s + (4.29 − 0.725i)19-s + ⋯
L(s)  = 1  + (0.938 + 0.165i)2-s + (1.06 − 1.27i)3-s + (−0.0868 − 0.0316i)4-s + (1.21 − 1.01i)6-s + (−0.113 + 0.0656i)7-s + (−0.901 − 0.520i)8-s + (−0.306 − 1.73i)9-s + (0.335 − 0.581i)11-s + (−0.133 + 0.0768i)12-s + (0.458 + 0.546i)13-s + (−0.117 + 0.0427i)14-s + (−0.688 − 0.577i)16-s + (−0.111 − 0.0197i)17-s − 1.68i·18-s + (0.986 − 0.166i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.262 + 0.964i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.262 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16101 - 1.65109i\)
\(L(\frac12)\) \(\approx\) \(2.16101 - 1.65109i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.29 + 0.725i)T \)
good2 \( 1 + (-1.32 - 0.233i)T + (1.87 + 0.684i)T^{2} \)
3 \( 1 + (-1.85 + 2.20i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.300 - 0.173i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.11 + 1.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.65 - 1.97i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.460 + 0.0812i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.921 - 2.53i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.19 - 6.77i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.55 - 6.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.94iT - 37T^{2} \)
41 \( 1 + (-1.89 - 1.59i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.33 + 3.66i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (7.18 - 1.26i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.970 - 2.66i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.09 + 6.20i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (8.57 + 3.12i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-7.55 + 1.33i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-8.74 + 3.18i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.892 - 1.06i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-9.07 - 7.61i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (12.8 - 7.41i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.88 + 6.61i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (9.30 + 1.64i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.15780882054459471087599136906, −9.533325472630842160335210105791, −8.927124866028311860696880047387, −8.067332646608633700857648723701, −6.93625168520221439458236118164, −6.32546837583617993913936534164, −5.14239497418449881381514643775, −3.68839765368486101568749814260, −2.94922720973595386765095339685, −1.31931840680458621686843238919, 2.56457895591992553472076571191, 3.49693285505752815392716043980, 4.26733808268712384983498143804, 5.04630314416266416884932354421, 6.23252622993782703461503250352, 7.894263154457145511073609532839, 8.571845775871462562004637377666, 9.605034969436739986623258221057, 10.00255839823173532380931084717, 11.26073372015174340110771620225

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