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Properties

Label 2-475-95.4-c1-0-20
Degree $2$
Conductor $475$
Sign $-0.924 + 0.380i$
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  + (0.575 − 1.58i)2-s + (−3.20 + 0.564i)3-s + (−0.639 − 0.536i)4-s + (−0.950 + 5.39i)6-s + (0.474 + 0.274i)7-s + (1.69 − 0.981i)8-s + (7.11 − 2.59i)9-s + (−0.165 − 0.286i)11-s + (2.35 + 1.35i)12-s + (−4.74 − 0.837i)13-s + (0.707 − 0.593i)14-s + (−0.863 − 4.89i)16-s + (1.80 − 4.96i)17-s − 12.7i·18-s + (−4.30 − 0.670i)19-s + ⋯
L(s)  = 1  + (0.407 − 1.11i)2-s + (−1.84 + 0.326i)3-s + (−0.319 − 0.268i)4-s + (−0.388 + 2.20i)6-s + (0.179 + 0.103i)7-s + (0.600 − 0.346i)8-s + (2.37 − 0.863i)9-s + (−0.0499 − 0.0864i)11-s + (0.678 + 0.391i)12-s + (−1.31 − 0.232i)13-s + (0.189 − 0.158i)14-s + (−0.215 − 1.22i)16-s + (0.438 − 1.20i)17-s − 3.00i·18-s + (−0.988 − 0.153i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.924 + 0.380i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.924 + 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143759 - 0.727551i\)
\(L(\frac12)\) \(\approx\) \(0.143759 - 0.727551i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.30 + 0.670i)T \)
good2 \( 1 + (-0.575 + 1.58i)T + (-1.53 - 1.28i)T^{2} \)
3 \( 1 + (3.20 - 0.564i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.474 - 0.274i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.165 + 0.286i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.74 + 0.837i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.80 + 4.96i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (0.713 - 0.850i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.01 - 1.09i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.01 + 5.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.67iT - 37T^{2} \)
41 \( 1 + (1.37 + 7.79i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.05 + 1.25i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.57 - 4.32i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.32 + 5.15i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (6.39 + 2.32i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.520 - 0.436i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.65 - 7.30i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.832 - 0.698i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-13.7 + 2.42i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.243 - 1.38i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.740 + 0.427i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.52 - 14.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.40 - 6.59i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.91792846172885075084128161643, −10.10937221572523859638766166112, −9.496956945566744445425948301102, −7.55773910593422031002855644720, −6.78533529695726826981906175407, −5.48191230519940208399234595637, −4.85041854939926012390739088146, −3.88774671969551189066135418293, −2.24627360678378287960890522445, −0.49886753640935305665526128439, 1.68665999891264526829625386365, 4.40970082038941544769580134021, 4.98173664737938828187345302080, 5.97620330479131768072715483370, 6.52873475829630621044330456276, 7.34875456412940501501347076156, 8.188350292396647927821576439615, 9.986271619667338613427661369572, 10.59418012379706688293615493426, 11.43763875397530541375200373065

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