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Properties

Label 2-475-95.24-c1-0-14
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: $\chi_{475} (24, \cdot )$
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

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

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