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Properties

Label 2-475-475.6-c1-0-10
Degree $2$
Conductor $475$
Sign $0.925 - 0.379i$
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  + (−0.954 − 0.596i)2-s + (−0.197 + 0.0567i)3-s + (−0.321 − 0.659i)4-s + (−2.23 + 0.136i)5-s + (0.222 + 0.0638i)6-s + (0.295 − 0.511i)7-s + (−0.321 + 3.05i)8-s + (−2.50 + 1.56i)9-s + (2.21 + 1.20i)10-s + (2.24 − 2.49i)11-s + (0.101 + 0.112i)12-s + (0.122 + 3.52i)13-s + (−0.586 + 0.312i)14-s + (0.433 − 0.153i)15-s + (1.22 − 1.57i)16-s + (3.49 − 0.244i)17-s + ⋯
L(s)  = 1  + (−0.674 − 0.421i)2-s + (−0.114 + 0.0327i)3-s + (−0.160 − 0.329i)4-s + (−0.998 + 0.0609i)5-s + (0.0908 + 0.0260i)6-s + (0.111 − 0.193i)7-s + (−0.113 + 1.08i)8-s + (−0.836 + 0.522i)9-s + (0.699 + 0.379i)10-s + (0.676 − 0.751i)11-s + (0.0291 + 0.0323i)12-s + (0.0341 + 0.976i)13-s + (−0.156 + 0.0834i)14-s + (0.112 − 0.0396i)15-s + (0.306 − 0.392i)16-s + (0.847 − 0.0592i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.925 - 0.379i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583190 + 0.115073i\)
\(L(\frac12)\) \(\approx\) \(0.583190 + 0.115073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 - 0.136i)T \)
19 \( 1 + (3.40 - 2.71i)T \)
good2 \( 1 + (0.954 + 0.596i)T + (0.876 + 1.79i)T^{2} \)
3 \( 1 + (0.197 - 0.0567i)T + (2.54 - 1.58i)T^{2} \)
7 \( 1 + (-0.295 + 0.511i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.24 + 2.49i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.122 - 3.52i)T + (-12.9 + 0.906i)T^{2} \)
17 \( 1 + (-3.49 + 0.244i)T + (16.8 - 2.36i)T^{2} \)
23 \( 1 + (-5.85 + 0.823i)T + (22.1 - 6.33i)T^{2} \)
29 \( 1 + (-9.33 - 0.652i)T + (28.7 + 4.03i)T^{2} \)
31 \( 1 + (8.83 - 3.93i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-0.775 + 2.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (4.01 - 5.13i)T + (-9.91 - 39.7i)T^{2} \)
43 \( 1 + (-0.376 - 2.13i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.08 - 0.285i)T + (46.5 + 6.54i)T^{2} \)
53 \( 1 + (-5.57 - 11.4i)T + (-32.6 + 41.7i)T^{2} \)
59 \( 1 + (-0.979 - 2.42i)T + (-42.4 + 40.9i)T^{2} \)
61 \( 1 + (-1.41 + 0.198i)T + (58.6 - 16.8i)T^{2} \)
67 \( 1 + (0.446 + 1.79i)T + (-59.1 + 31.4i)T^{2} \)
71 \( 1 + (5.43 + 5.25i)T + (2.47 + 70.9i)T^{2} \)
73 \( 1 + (0.220 - 6.31i)T + (-72.8 - 5.09i)T^{2} \)
79 \( 1 + (-0.559 + 0.160i)T + (66.9 - 41.8i)T^{2} \)
83 \( 1 + (2.47 - 1.10i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-0.114 - 0.146i)T + (-21.5 + 86.3i)T^{2} \)
97 \( 1 + (2.06 - 8.27i)T + (-85.6 - 45.5i)T^{2} \)
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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

−10.95915170763345851703139649858, −10.44428921879204214470784458390, −9.025369747212990337852069000991, −8.658592745645744649036504913819, −7.69912377647282138479438078118, −6.47420421378422398016625924710, −5.33377184454859741355624882937, −4.23858649664551254286740688702, −2.90090894819923200542680616339, −1.15643640799866172833393039329, 0.57200537043240906651115554729, 3.07321064336179634623736054739, 4.02041611768965823525356824490, 5.30320245558445652601722021446, 6.70514365148715623355929007758, 7.35825640654818246618000815522, 8.428095244575567687042281252670, 8.795547338839245608890454696690, 9.872929404353030517429773102683, 10.95051738022781370972375625592

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