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Properties

Label 2-475-95.4-c1-0-26
Degree $2$
Conductor $475$
Sign $-0.591 - 0.806i$
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  + (0.883 − 2.42i)2-s + (−0.243 + 0.0430i)3-s + (−3.57 − 3.00i)4-s + (−0.111 + 0.630i)6-s + (0.347 + 0.200i)7-s + (−5.97 + 3.45i)8-s + (−2.76 + 1.00i)9-s + (−2.59 − 4.49i)11-s + (1.00 + 0.578i)12-s + (−2.84 − 0.501i)13-s + (0.794 − 0.666i)14-s + (1.47 + 8.35i)16-s + (−1.41 + 3.89i)17-s + 7.59i·18-s + (−0.386 − 4.34i)19-s + ⋯
L(s)  = 1  + (0.624 − 1.71i)2-s + (−0.140 + 0.0248i)3-s + (−1.78 − 1.50i)4-s + (−0.0453 + 0.257i)6-s + (0.131 + 0.0759i)7-s + (−2.11 + 1.22i)8-s + (−0.920 + 0.335i)9-s + (−0.782 − 1.35i)11-s + (0.289 + 0.167i)12-s + (−0.789 − 0.139i)13-s + (0.212 − 0.178i)14-s + (0.368 + 2.08i)16-s + (−0.343 + 0.944i)17-s + 1.78i·18-s + (−0.0887 − 0.996i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.591 - 0.806i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448924 + 0.886630i\)
\(L(\frac12)\) \(\approx\) \(0.448924 + 0.886630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.386 + 4.34i)T \)
good2 \( 1 + (-0.883 + 2.42i)T + (-1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.243 - 0.0430i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.347 - 0.200i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 + 4.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.84 + 0.501i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.41 - 3.89i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.15 + 2.57i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-6.18 + 2.25i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.13 + 5.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.14iT - 37T^{2} \)
41 \( 1 + (-0.496 - 2.81i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.99 - 9.52i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.31 + 6.35i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-7.90 + 9.42i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (1.42 + 0.518i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.35 + 4.49i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.258 - 0.711i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.38 - 5.35i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (9.76 - 1.72i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.553 + 3.13i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.77 - 2.75i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.17 + 12.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (3.04 - 8.35i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.73710523884880029360891656194, −9.983505104507000460434781029908, −8.816083245786241472244681166735, −8.137856845087831427216493843989, −6.22751638215418647776727272004, −5.28864110448161957186589816939, −4.46579104964399197730311180826, −3.04982448234926710116842889640, −2.43018969152734050771513252944, −0.48722851020316501835462507472, 2.85888287907790841739192191687, 4.41960774003109453051258750627, 5.10424454053299297062783265520, 5.97666321578104176951603590509, 7.10088748426684371566548988979, 7.54185628112491476806614815021, 8.618995541485281673362626870952, 9.439643251377607930572844645978, 10.58494844367408858607254421692, 12.18496568499895640636934902632

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