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Properties

Label 2-475-19.5-c1-0-8
Degree $2$
Conductor $475$
Sign $-0.706 - 0.707i$
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  + (1.18 − 0.429i)2-s + (−0.523 + 2.96i)3-s + (−0.321 + 0.270i)4-s + (0.657 + 3.72i)6-s + (1.86 + 3.22i)7-s + (−1.52 + 2.63i)8-s + (−5.70 − 2.07i)9-s + (1.67 − 2.90i)11-s + (−0.632 − 1.09i)12-s + (−0.840 − 4.76i)13-s + (3.58 + 3.00i)14-s + (−0.518 + 2.93i)16-s + (−2.51 + 0.914i)17-s − 7.63·18-s + (0.961 + 4.25i)19-s + ⋯
L(s)  = 1  + (0.835 − 0.303i)2-s + (−0.301 + 1.71i)3-s + (−0.160 + 0.135i)4-s + (0.268 + 1.52i)6-s + (0.703 + 1.21i)7-s + (−0.537 + 0.931i)8-s + (−1.90 − 0.692i)9-s + (0.505 − 0.876i)11-s + (−0.182 − 0.316i)12-s + (−0.233 − 1.32i)13-s + (0.958 + 0.804i)14-s + (−0.129 + 0.734i)16-s + (−0.609 + 0.221i)17-s − 1.79·18-s + (0.220 + 0.975i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.706 - 0.707i$
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: \((2,\ 475,\ (\ :1/2),\ -0.706 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630640 + 1.52177i\)
\(L(\frac12)\) \(\approx\) \(0.630640 + 1.52177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.961 - 4.25i)T \)
good2 \( 1 + (-1.18 + 0.429i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.523 - 2.96i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-1.86 - 3.22i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.67 + 2.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.840 + 4.76i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.51 - 0.914i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.43 - 1.20i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.93 - 1.79i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.55 - 2.70i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.992T + 37T^{2} \)
41 \( 1 + (-0.0723 + 0.410i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.52 - 4.64i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.10 - 0.766i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.199 - 0.167i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-4.87 + 1.77i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.589 + 0.494i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-10.1 - 3.68i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.53 - 1.28i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.792 - 4.49i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (2.09 - 11.8i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (6.78 + 11.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.33 + 7.55i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (6.79 - 2.47i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.39688620519593334484290668939, −10.64255241582270538407276136752, −9.650497258510142501444648677912, −8.630047489693172888281642559922, −8.290847316971069085613482420274, −5.90350020667524024170923758582, −5.47036653635094207828144505409, −4.62595491886341719014034422035, −3.63719255636572424045204073610, −2.76425044304747317668223343897, 0.841655511667984053854185386642, 2.15611299589223434089874806721, 4.17630871112662267388890870675, 4.86511613768710504851050406911, 6.25696245216060028862931590253, 6.95710762369598687880757505673, 7.35646949944095327994857836698, 8.662015108003607620262504041090, 9.761903184003163085011335172334, 11.07383345247217529045776281644

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