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Properties

Label 2-475-95.12-c1-0-11
Degree $2$
Conductor $475$
Sign $-0.240 + 0.970i$
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  + (−1.36 − 0.366i)2-s + (−0.732 + 2.73i)3-s + (2 − 3.46i)6-s + (1.99 + 2i)8-s + (−4.33 − 2.5i)9-s − 3·11-s + (−4.09 + 1.09i)13-s + (−1.99 − 3.46i)16-s + (−0.633 + 2.36i)17-s + (4.99 + 5i)18-s + (4.33 + 0.5i)19-s + (4.09 + 1.09i)22-s + (−1.90 − 7.09i)23-s + (−6.92 + 3.99i)24-s + 6·26-s + (4 − 3.99i)27-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.422 + 1.57i)3-s + (0.816 − 1.41i)6-s + (0.707 + 0.707i)8-s + (−1.44 − 0.833i)9-s − 0.904·11-s + (−1.13 + 0.304i)13-s + (−0.499 − 0.866i)16-s + (−0.153 + 0.573i)17-s + (1.17 + 1.17i)18-s + (0.993 + 0.114i)19-s + (0.873 + 0.234i)22-s + (−0.396 − 1.48i)23-s + (−1.41 + 0.816i)24-s + 1.17·26-s + (0.769 − 0.769i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.240 + 0.970i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0435050 - 0.0555968i\)
\(L(\frac12)\) \(\approx\) \(0.0435050 - 0.0555968i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 + 0.366i)T + (1.73 + i)T^{2} \)
3 \( 1 + (0.732 - 2.73i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (4.09 - 1.09i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.633 - 2.36i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (1.90 + 7.09i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.59 - 4.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (-9 + 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.09 + 1.90i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.36 - 0.633i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.73 + 0.732i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.29 + 12.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (13.5 - 7.79i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.09 + 1.90i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.59 - 4.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.66 - 8.66i)T - 83iT^{2} \)
89 \( 1 + (2.59 - 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.2 - 3.29i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.49890622768486955247571859765, −9.895538272230871373242987881493, −9.315410871316748088873759993577, −8.390398909767534504175608476360, −7.38834408729766685730826869453, −5.70868296278435758549100135844, −4.94110891645519155881238027899, −4.07982242160045680463555309820, −2.45774135153492262784082387861, −0.06586822048355023226338082860, 1.35783394489646359915014011651, 2.85171147899250150947531296447, 4.87958414139656418601667702748, 5.92597004056412450748458987317, 7.27569475708010640626973395882, 7.44489232298719777154261504515, 8.235913500356003523149797321690, 9.432456831175801769910605258797, 10.16629010660443776472243089964, 11.40367928458893199962634943057

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