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Properties

Label 2-475-95.54-c1-0-2
Degree $2$
Conductor $475$
Sign $0.116 + 0.993i$
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  + (−1.47 + 1.75i)2-s + (1.13 + 3.10i)3-s + (−0.563 − 3.19i)4-s + (−7.11 − 2.59i)6-s + (−2.54 + 1.46i)7-s + (2.46 + 1.42i)8-s + (−6.08 + 5.10i)9-s + (0.288 − 0.500i)11-s + (9.29 − 5.36i)12-s + (0.229 − 0.629i)13-s + (1.16 − 6.62i)14-s + (−0.0331 + 0.0120i)16-s + (0.226 − 0.269i)17-s − 18.1i·18-s + (3.86 + 2.02i)19-s + ⋯
L(s)  = 1  + (−1.04 + 1.24i)2-s + (0.653 + 1.79i)3-s + (−0.281 − 1.59i)4-s + (−2.90 − 1.05i)6-s + (−0.962 + 0.555i)7-s + (0.872 + 0.503i)8-s + (−2.02 + 1.70i)9-s + (0.0870 − 0.150i)11-s + (2.68 − 1.54i)12-s + (0.0635 − 0.174i)13-s + (0.312 − 1.77i)14-s + (−0.00829 + 0.00302i)16-s + (0.0548 − 0.0653i)17-s − 4.28i·18-s + (0.886 + 0.463i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430763 - 0.383211i\)
\(L(\frac12)\) \(\approx\) \(0.430763 - 0.383211i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.86 - 2.02i)T \)
good2 \( 1 + (1.47 - 1.75i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (-1.13 - 3.10i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (2.54 - 1.46i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.288 + 0.500i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.229 + 0.629i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.226 + 0.269i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (4.05 - 0.715i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (2.30 - 1.93i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.148 + 0.257i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.30iT - 37T^{2} \)
41 \( 1 + (2.51 - 0.913i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-6.49 - 1.14i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-7.09 - 8.45i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (4.04 - 0.713i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.467 + 0.392i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.178 - 1.01i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.14 + 2.55i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.29 - 13.0i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-2.44 - 6.70i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-1.44 + 0.527i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (11.5 - 6.65i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.17 + 2.24i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (7.84 - 9.34i)T + (-16.8 - 95.5i)T^{2} \)
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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

−11.15254208180241998746742934924, −10.20920819173964355258632057769, −9.560312201113957591555733937300, −9.158789141204442803904950503124, −8.330230277066284001543831081561, −7.44626295713557330595730219176, −5.96721032478697068736389867274, −5.45112425008769462770658558993, −3.98603707515884376700217687020, −2.90984957372443607136987680507, 0.45180546908834671349072101823, 1.66676650220220515341098352162, 2.75589502544088923197032865750, 3.61352076864882052060993694390, 6.04313365661831557257060490902, 7.03175595830731051641560892082, 7.73854380456924613025091442026, 8.627202765761469093953732510437, 9.392834389800035584109682505841, 10.18884378700991034587433645313

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