Properties

Label 2-475-19.5-c1-0-3
Degree $2$
Conductor $475$
Sign $-0.999 + 0.0359i$
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  + (−0.234 + 0.0854i)2-s + (−0.399 + 2.26i)3-s + (−1.48 + 1.24i)4-s + (−0.0998 − 0.566i)6-s + (1.98 + 3.42i)7-s + (0.491 − 0.851i)8-s + (−2.15 − 0.785i)9-s + (−1.56 + 2.71i)11-s + (−2.22 − 3.86i)12-s + (0.461 + 2.61i)13-s + (−0.757 − 0.635i)14-s + (0.630 − 3.57i)16-s + (2.12 − 0.771i)17-s + 0.573·18-s + (−3.76 − 2.19i)19-s + ⋯
L(s)  = 1  + (−0.165 + 0.0604i)2-s + (−0.230 + 1.30i)3-s + (−0.742 + 0.622i)4-s + (−0.0407 − 0.231i)6-s + (0.748 + 1.29i)7-s + (0.173 − 0.301i)8-s + (−0.719 − 0.261i)9-s + (−0.472 + 0.818i)11-s + (−0.643 − 1.11i)12-s + (0.128 + 0.726i)13-s + (−0.202 − 0.169i)14-s + (0.157 − 0.893i)16-s + (0.514 − 0.187i)17-s + 0.135·18-s + (−0.864 − 0.502i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.999 + 0.0359i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.999 + 0.0359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0166596 - 0.926715i\)
\(L(\frac12)\) \(\approx\) \(0.0166596 - 0.926715i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.76 + 2.19i)T \)
good2 \( 1 + (0.234 - 0.0854i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.399 - 2.26i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-1.98 - 3.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.56 - 2.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.461 - 2.61i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-2.12 + 0.771i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.81 + 4.87i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.28 + 0.466i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.447 + 0.774i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.62T + 37T^{2} \)
41 \( 1 + (-1.08 + 6.14i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.35 - 1.13i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.165 - 0.0603i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.19 - 4.35i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-7.34 + 2.67i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.796 - 0.668i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-14.2 - 5.20i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.99 - 6.70i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.726 - 4.11i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.94 + 11.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-6.62 - 11.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.257 - 1.46i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (14.4 - 5.26i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.33126580148993042951694451671, −10.49157020385561234397361314832, −9.472150031462229559285824036548, −8.955028751021140285554051208711, −8.209375296886093130959933183228, −6.94668338985421540927493282719, −5.30327558389759296234680594172, −4.84113757833838007062242093794, −3.91929370336331499762016854345, −2.45169912899588953393931673197, 0.67428124087338899081659004718, 1.59710750026121920572451854721, 3.61325467066321765415079527525, 4.99135609254063845860635259715, 5.86534245804165982361626392742, 6.97643322470069312471309826568, 7.910290243458088467147480955246, 8.402571373519563425216608753637, 9.795523976299238767428059328728, 10.75065387649719988094757796540

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