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Properties

Label 2-475-19.6-c1-0-20
Degree $2$
Conductor $475$
Sign $-0.0591 + 0.998i$
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  + (−0.528 + 0.443i)2-s + (−0.652 − 0.237i)3-s + (−0.264 + 1.50i)4-s + (0.450 − 0.164i)6-s + (−1.16 + 2.02i)7-s + (−1.21 − 2.10i)8-s + (−1.92 − 1.61i)9-s + (−2.28 − 3.96i)11-s + (0.529 − 0.916i)12-s + (1.20 − 0.438i)13-s + (−0.279 − 1.58i)14-s + (−1.28 − 0.468i)16-s + (0.501 − 0.420i)17-s + 1.73·18-s + (3.67 + 2.34i)19-s + ⋯
L(s)  = 1  + (−0.373 + 0.313i)2-s + (−0.376 − 0.137i)3-s + (−0.132 + 0.750i)4-s + (0.183 − 0.0669i)6-s + (−0.441 + 0.764i)7-s + (−0.429 − 0.744i)8-s + (−0.642 − 0.539i)9-s + (−0.690 − 1.19i)11-s + (0.152 − 0.264i)12-s + (0.333 − 0.121i)13-s + (−0.0747 − 0.424i)14-s + (−0.321 − 0.117i)16-s + (0.121 − 0.102i)17-s + 0.409·18-s + (0.843 + 0.537i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0591 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0591 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.0591 + 0.998i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.0591 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235233 - 0.249594i\)
\(L(\frac12)\) \(\approx\) \(0.235233 - 0.249594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.67 - 2.34i)T \)
good2 \( 1 + (0.528 - 0.443i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (0.652 + 0.237i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (1.16 - 2.02i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.28 + 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.20 + 0.438i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.501 + 0.420i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.966 + 5.48i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.62 + 3.04i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.24 + 3.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + (8.17 + 2.97i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.66 + 9.44i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.84 + 4.06i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.14 - 6.50i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-4.51 + 3.78i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.30 - 7.38i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (10.0 + 8.39i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.651 - 3.69i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-7.48 - 2.72i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (5.92 + 2.15i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (4.91 - 8.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.4 - 4.16i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (3.22 - 2.70i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.84016015794262316643624941251, −9.700820555989170413994706888609, −8.674771362580425436080780305922, −8.347000054649082282878566586986, −7.10161211371771612797629492816, −6.10397366021933540061003071803, −5.41243412150328266750633963817, −3.62881682960399952800962967641, −2.82892543002673284302195346855, −0.24544293622944518923946890249, 1.59845897014362309644838130530, 3.14427152852503914820605123732, 4.80980088080907290747292290521, 5.39183354292370656402248989853, 6.63738442301138647487185272070, 7.59110018466646710822168929370, 8.737071377321982602057840313719, 9.813173258225977116758701140482, 10.21886143048117008395021186412, 11.11495408921212497846498578044

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