Properties

Label 2-475-95.88-c1-0-15
Degree $2$
Conductor $475$
Sign $0.141 - 0.989i$
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  + (−0.258 + 0.965i)2-s + (3.08 + 0.826i)3-s + (0.866 + 0.5i)4-s + (−1.59 + 2.76i)6-s + (1.22 − 1.22i)7-s + (−2.12 + 2.12i)8-s + (6.22 + 3.59i)9-s − 4.19·11-s + (2.25 + 2.25i)12-s + (−1.08 − 4.04i)13-s + (0.866 + 1.49i)14-s + (−0.500 − 0.866i)16-s + (−5.34 − 1.43i)17-s + (−5.08 + 5.08i)18-s + (4.33 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (1.78 + 0.477i)3-s + (0.433 + 0.250i)4-s + (−0.651 + 1.12i)6-s + (0.462 − 0.462i)7-s + (−0.749 + 0.749i)8-s + (2.07 + 1.19i)9-s − 1.26·11-s + (0.651 + 0.651i)12-s + (−0.300 − 1.12i)13-s + (0.231 + 0.400i)14-s + (−0.125 − 0.216i)16-s + (−1.29 − 0.347i)17-s + (−1.19 + 1.19i)18-s + (0.993 + 0.114i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.141 - 0.989i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.141 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86740 + 1.61917i\)
\(L(\frac12)\) \(\approx\) \(1.86740 + 1.61917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.33 - 0.5i)T \)
good2 \( 1 + (0.258 - 0.965i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-3.08 - 0.826i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \)
11 \( 1 + 4.19T + 11T^{2} \)
13 \( 1 + (1.08 + 4.04i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (5.34 + 1.43i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (1.67 - 0.448i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.76 + 4.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.06iT - 31T^{2} \)
37 \( 1 + (1.68 + 1.68i)T + 37iT^{2} \)
41 \( 1 + (-3.28 + 1.89i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0863 - 0.322i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.41 + 9.00i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.81 - 6.76i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.66 - 8.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.28 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 - 2.78i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.07 - 1.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.22 + 12.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.69 - 9.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.13 - 5.13i)T + 83iT^{2} \)
89 \( 1 + (2.59 - 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.86 - 6.94i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.86701982936494193234024400801, −10.18072577220674109938309264176, −9.149084328869005695941079620844, −8.255955103261218060414392368822, −7.73941969397977765408892709955, −7.18457268258838947613259087064, −5.52847169625967660939682923694, −4.34763764019061712203061833386, −3.01563541055779342985308288811, −2.34776623076564939392420448715, 1.72686884662709735314913321897, 2.42242661169991062698614901417, 3.33932075519819612712407857503, 4.78624197058368857646399035956, 6.48801983056209206267863658276, 7.33320175554072712143644319833, 8.268068661809258952052593625368, 9.060526202133723945476012268131, 9.741737804727788779781349352823, 10.72994659206444823591589427293

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