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Properties

Label 2-475-95.4-c1-0-5
Degree $2$
Conductor $475$
Sign $-0.948 - 0.315i$
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  + (−0.429 + 1.18i)2-s + (−2.96 + 0.523i)3-s + (0.321 + 0.270i)4-s + (0.657 − 3.72i)6-s + (3.22 + 1.86i)7-s + (−2.63 + 1.52i)8-s + (5.70 − 2.07i)9-s + (1.67 + 2.90i)11-s + (−1.09 − 0.632i)12-s + (4.76 + 0.840i)13-s + (−3.58 + 3.00i)14-s + (−0.518 − 2.93i)16-s + (0.914 − 2.51i)17-s + 7.63i·18-s + (−0.961 + 4.25i)19-s + ⋯
L(s)  = 1  + (−0.303 + 0.835i)2-s + (−1.71 + 0.301i)3-s + (0.160 + 0.135i)4-s + (0.268 − 1.52i)6-s + (1.21 + 0.703i)7-s + (−0.931 + 0.537i)8-s + (1.90 − 0.692i)9-s + (0.505 + 0.876i)11-s + (−0.316 − 0.182i)12-s + (1.32 + 0.233i)13-s + (−0.958 + 0.804i)14-s + (−0.129 − 0.734i)16-s + (0.221 − 0.609i)17-s + 1.79i·18-s + (−0.220 + 0.975i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.948 - 0.315i$
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: \((2,\ 475,\ (\ :1/2),\ -0.948 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133663 + 0.824551i\)
\(L(\frac12)\) \(\approx\) \(0.133663 + 0.824551i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.429 - 1.18i)T + (-1.53 - 1.28i)T^{2} \)
3 \( 1 + (2.96 - 0.523i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-3.22 - 1.86i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.67 - 2.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.76 - 0.840i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.914 + 2.51i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.20 - 1.43i)T + (-3.99 - 22.6i)T^{2} \)
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 \( 1 + 0.992iT - 37T^{2} \)
41 \( 1 + (-0.0723 - 0.410i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.64 + 5.52i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.766 - 2.10i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.167 - 0.199i)T + (-9.20 - 52.1i)T^{2} \)
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 \( 1 + (-3.68 - 10.1i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.53 + 1.28i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (4.49 - 0.792i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.09 - 11.8i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-11.7 - 6.78i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.33 + 7.55i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-2.47 + 6.79i)T + (-74.3 - 62.3i)T^{2} \)
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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

−11.56572400611583197622608707647, −10.77158102462804016757375799840, −9.599327035447989287428625045958, −8.571893782893124548914389805900, −7.59190448793911577407160605203, −6.64132979731889190054076400850, −5.82868239346735755330992553049, −5.20471505807508521410960858974, −4.01162727881016858650944704839, −1.68852810233871659860975278811, 0.77919001435379466673235427918, 1.60886490200961710172882212311, 3.71415694959595473693668250236, 4.93787058696871846138921868028, 6.05049658254405014591485255575, 6.55172751988808926013370050991, 7.83947230771389231617189096868, 8.975371034471211158915301045144, 10.40976533306803029619367502967, 10.79615171129111448251700951693

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