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Properties

Label 2-475-95.9-c1-0-10
Degree $2$
Conductor $475$
Sign $0.605 - 0.795i$
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  + (−1.44 + 0.253i)2-s + (0.970 + 1.15i)3-s + (0.130 − 0.0473i)4-s + (−1.69 − 1.41i)6-s + (3.52 + 2.03i)7-s + (2.35 − 1.36i)8-s + (0.124 − 0.707i)9-s + (0.310 + 0.537i)11-s + (0.180 + 0.104i)12-s + (3.27 − 3.90i)13-s + (−5.59 − 2.03i)14-s + (−3.26 + 2.73i)16-s + (−0.262 + 0.0462i)17-s + 1.05i·18-s + (−0.399 − 4.34i)19-s + ⋯
L(s)  = 1  + (−1.01 + 0.179i)2-s + (0.560 + 0.667i)3-s + (0.0650 − 0.0236i)4-s + (−0.690 − 0.579i)6-s + (1.33 + 0.769i)7-s + (0.833 − 0.481i)8-s + (0.0416 − 0.235i)9-s + (0.0936 + 0.162i)11-s + (0.0522 + 0.0301i)12-s + (0.908 − 1.08i)13-s + (−1.49 − 0.544i)14-s + (−0.815 + 0.684i)16-s + (−0.0636 + 0.0112i)17-s + 0.247i·18-s + (−0.0917 − 0.995i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02122 + 0.506221i\)
\(L(\frac12)\) \(\approx\) \(1.02122 + 0.506221i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.399 + 4.34i)T \)
good2 \( 1 + (1.44 - 0.253i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (-0.970 - 1.15i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-3.52 - 2.03i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.310 - 0.537i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.27 + 3.90i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.262 - 0.0462i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.99 - 5.48i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.708 - 4.01i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.83iT - 37T^{2} \)
41 \( 1 + (3.43 - 2.88i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.615 - 1.69i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (11.3 + 2.00i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.862 - 2.37i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.154 + 0.876i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.03 - 0.742i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-13.8 - 2.44i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.54 - 0.563i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.15 - 1.37i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (3.94 - 3.30i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (11.9 + 6.89i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.000572 - 0.000480i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (10.2 - 1.80i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.96667613950155313913531594522, −9.923631863575132743784884297606, −9.273559502678983830512600713947, −8.337684768526884003473889353982, −8.164037784108438285844716641752, −6.80033791352668853744489070356, −5.33002806424835824922440095886, −4.39828452526406017787643223218, −3.10745040550242497332128995346, −1.35366917161664333593532876279, 1.25337538167590615402965319136, 2.04290950326802408643939236945, 4.05007548963332043791696047721, 5.01229667239770667742483425554, 6.67247601775462664594546533678, 7.60095821226481460367242353427, 8.347840504443000950536628432238, 8.714113003518653269866605035008, 9.986195092079022616736084455385, 10.85772059500807010108451432267

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