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Properties

Label 2-475-19.16-c1-0-28
Degree $2$
Conductor $475$
Sign $-0.402 - 0.915i$
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  + (−1.24 − 1.04i)2-s + (1.11 − 0.406i)3-s + (0.107 + 0.611i)4-s + (−1.80 − 0.658i)6-s + (−1.11 − 1.93i)7-s + (−1.11 + 1.93i)8-s + (−1.21 + 1.02i)9-s + (−2.82 + 4.88i)11-s + (0.369 + 0.639i)12-s + (−4.41 − 1.60i)13-s + (−0.627 + 3.55i)14-s + (4.56 − 1.66i)16-s + (−0.601 − 0.505i)17-s + 2.56·18-s + (−2.63 + 3.47i)19-s + ⋯
L(s)  = 1  + (−0.877 − 0.735i)2-s + (0.644 − 0.234i)3-s + (0.0539 + 0.305i)4-s + (−0.738 − 0.268i)6-s + (−0.421 − 0.730i)7-s + (−0.394 + 0.683i)8-s + (−0.405 + 0.340i)9-s + (−0.850 + 1.47i)11-s + (0.106 + 0.184i)12-s + (−1.22 − 0.445i)13-s + (−0.167 + 0.951i)14-s + (1.14 − 0.415i)16-s + (−0.146 − 0.122i)17-s + 0.605·18-s + (−0.603 + 0.797i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.402 - 0.915i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.402 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00786068 + 0.0120478i\)
\(L(\frac12)\) \(\approx\) \(0.00786068 + 0.0120478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 3.47i)T \)
good2 \( 1 + (1.24 + 1.04i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (-1.11 + 0.406i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.11 + 1.93i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.82 - 4.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.41 + 1.60i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.601 + 0.505i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.890 + 5.05i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.32 + 1.95i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.05 + 7.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.985T + 37T^{2} \)
41 \( 1 + (1.33 - 0.484i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.264 - 1.49i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.735 + 0.617i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.766 - 4.34i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-7.56 - 6.34i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.363 - 2.06i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.48 - 2.08i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.24 + 7.06i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (14.2 - 5.18i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.25 - 0.458i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3.76 - 6.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.19 + 1.16i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (9.77 + 8.20i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.08510443941557740188087797758, −9.877618964242335895167755951990, −8.667617482714259890523423581414, −7.83284457865539889851069766820, −7.17390335775243577051925733113, −5.60426270786703316246257615856, −4.40547290894792965294817928199, −2.73243567749024542638619911227, −2.07102098281130484671072162879, −0.009789207450298655278307301671, 2.66923606997105847074117511042, 3.52485459721000771083402820662, 5.28425340196652118688094997567, 6.26411276922266927536908126826, 7.24311335337194763154664582183, 8.277729100591155862028957056743, 8.845824076263146988188170400841, 9.392232535305852256117822308702, 10.38164120648581837863870225101, 11.55179645632105496536585674878

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