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Properties

Label 2-475-95.44-c1-0-24
Degree $2$
Conductor $475$
Sign $0.521 + 0.853i$
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  + (1.00 + 1.20i)2-s + (0.781 − 2.14i)3-s + (−0.0809 + 0.459i)4-s + (3.37 − 1.22i)6-s + (−3.47 − 2.00i)7-s + (2.08 − 1.20i)8-s + (−1.70 − 1.42i)9-s + (−1.38 − 2.39i)11-s + (0.922 + 0.532i)12-s + (−0.953 − 2.61i)13-s + (−1.09 − 6.20i)14-s + (4.43 + 1.61i)16-s + (2.31 + 2.76i)17-s − 3.48i·18-s + (−1.79 + 3.97i)19-s + ⋯
L(s)  = 1  + (0.713 + 0.850i)2-s + (0.451 − 1.23i)3-s + (−0.0404 + 0.229i)4-s + (1.37 − 0.501i)6-s + (−1.31 − 0.758i)7-s + (0.737 − 0.425i)8-s + (−0.567 − 0.475i)9-s + (−0.417 − 0.722i)11-s + (0.266 + 0.153i)12-s + (−0.264 − 0.726i)13-s + (−0.292 − 1.65i)14-s + (1.10 + 0.403i)16-s + (0.562 + 0.670i)17-s − 0.822i·18-s + (−0.410 + 0.911i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.521 + 0.853i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.521 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83351 - 1.02812i\)
\(L(\frac12)\) \(\approx\) \(1.83351 - 1.02812i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 - 3.97i)T \)
good2 \( 1 + (-1.00 - 1.20i)T + (-0.347 + 1.96i)T^{2} \)
3 \( 1 + (-0.781 + 2.14i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (3.47 + 2.00i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.38 + 2.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.953 + 2.61i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.31 - 2.76i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.34 - 0.237i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-7.28 - 6.11i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.776 - 1.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.51iT - 37T^{2} \)
41 \( 1 + (-6.21 - 2.26i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-6.15 + 1.08i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.85 + 6.97i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (3.88 + 0.684i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (2.76 - 2.32i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.30 - 7.37i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (9.36 - 11.1i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.576 + 3.26i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (3.42 - 9.40i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.82 + 0.666i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.40 + 0.809i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.5 - 4.19i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-7.45 - 8.87i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.59090709176005081671502294632, −10.20037628908328967093954008603, −8.692985351261252606696963058418, −7.67961442988209921060026433798, −7.15349225284940448935415460617, −6.25897510137036097793348110017, −5.60824820886222779529172369506, −4.00034445576419417667784889464, −2.91076880172974505140348136091, −1.04174660092930070537128564153, 2.56006202678697269731408124076, 3.07288941588577122933325399628, 4.31928788542062430274394197964, 4.87461428198346350522813973462, 6.28886134360694988381304481757, 7.56477099170191311089184522219, 8.920947013171522383695612168539, 9.598842677276188817039666947685, 10.18783873880723743760871040440, 11.16019379387680939799886703031

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