Properties

Label 2-475-19.9-c1-0-19
Degree $2$
Conductor $475$
Sign $0.980 - 0.196i$
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  + (0.233 + 1.32i)2-s + (2.20 − 1.85i)3-s + (0.173 − 0.0632i)4-s + (2.97 + 2.49i)6-s + (0.173 − 0.300i)7-s + (1.47 + 2.54i)8-s + (0.918 − 5.21i)9-s + (1.11 + 1.92i)11-s + (0.266 − 0.460i)12-s + (−1.97 − 1.65i)13-s + (0.439 + 0.160i)14-s + (−2.75 + 2.31i)16-s + (−0.0812 − 0.460i)17-s + 7.12·18-s + (−4.29 − 0.725i)19-s + ⋯
L(s)  = 1  + (0.165 + 0.938i)2-s + (1.27 − 1.06i)3-s + (0.0868 − 0.0316i)4-s + (1.21 + 1.01i)6-s + (0.0656 − 0.113i)7-s + (0.520 + 0.901i)8-s + (0.306 − 1.73i)9-s + (0.335 + 0.581i)11-s + (0.0768 − 0.133i)12-s + (−0.546 − 0.458i)13-s + (0.117 + 0.0427i)14-s + (−0.688 + 0.577i)16-s + (−0.0197 − 0.111i)17-s + 1.68·18-s + (−0.986 − 0.166i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52286 + 0.250200i\)
\(L(\frac12)\) \(\approx\) \(2.52286 + 0.250200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.29 + 0.725i)T \)
good2 \( 1 + (-0.233 - 1.32i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-2.20 + 1.85i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.11 - 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.97 + 1.65i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.0812 + 0.460i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (2.53 - 0.921i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.19 - 6.77i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.55 + 6.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 + (-1.89 + 1.59i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.66 - 1.33i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.26 + 7.18i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.66 - 0.970i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.09 + 6.20i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.57 - 3.12i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.33 - 7.55i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-8.74 - 3.18i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.06 - 0.892i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (9.07 - 7.61i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (7.41 - 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.88 + 6.61i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.64 + 9.30i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.07901414283873515925487377492, −9.905186657970124019834213910967, −8.800483303686486132809558061923, −8.093633964353712272793774788554, −7.28704751876426357434137443348, −6.81150933863962527122336649875, −5.69355676990710149952873214298, −4.28944071022019593347706152450, −2.72914099560980989324378931192, −1.74735993475854625287661543876, 2.00666306149461077068560333997, 2.94735651775692975138863861035, 3.90728225753667539987723410268, 4.61261024789613533317444798852, 6.32193020958413674056072785049, 7.63005383123063399856778524397, 8.575148930847026518752901352956, 9.354191523255115532148150674891, 10.18347542596908265281737418574, 10.76518462353441440806025815005

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