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Properties

Label 2-475-95.27-c1-0-10
Degree $2$
Conductor $475$
Sign $-0.941 + 0.336i$
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  + (−0.578 − 2.15i)2-s + (−2.52 + 0.677i)3-s + (−2.59 + 1.5i)4-s + (2.92 + 5.06i)6-s + (1.22 + 1.22i)7-s + (1.58 + 1.58i)8-s + (3.33 − 1.92i)9-s + 4.85·11-s + (5.55 − 5.55i)12-s + (1.25 − 4.68i)13-s + (1.93 − 3.35i)14-s + (−0.500 + 0.866i)16-s + (−4.38 + 1.17i)17-s + (−6.09 − 6.09i)18-s + (−4.33 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.409 − 1.52i)2-s + (−1.46 + 0.391i)3-s + (−1.29 + 0.750i)4-s + (1.19 + 2.06i)6-s + (0.462 + 0.462i)7-s + (0.559 + 0.559i)8-s + (1.11 − 0.642i)9-s + 1.46·11-s + (1.60 − 1.60i)12-s + (0.348 − 1.30i)13-s + (0.517 − 0.896i)14-s + (−0.125 + 0.216i)16-s + (−1.06 + 0.284i)17-s + (−1.43 − 1.43i)18-s + (−0.993 + 0.114i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.941 + 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0953394 - 0.549624i\)
\(L(\frac12)\) \(\approx\) \(0.0953394 - 0.549624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.578 + 2.15i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (2.52 - 0.677i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \)
11 \( 1 - 4.85T + 11T^{2} \)
13 \( 1 + (-1.25 + 4.68i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (4.38 - 1.17i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-5.01 - 1.34i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.26 + 3.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.1iT - 31T^{2} \)
37 \( 1 + (2.62 - 2.62i)T - 37iT^{2} \)
41 \( 1 + (-0.572 - 0.330i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.17 + 8.12i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.00 + 3.74i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.17 - 4.37i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.25 + 5.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.28 + 9.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.79 - 1.55i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.35 + 1.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.0654 - 0.244i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.992 - 1.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.81 + 7.81i)T - 83iT^{2} \)
89 \( 1 + (-6.47 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.99 + 11.1i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.92152503712711212232603299918, −10.07940610504260498734477605041, −9.177923916178676829974416795209, −8.345580512099122042383051588167, −6.62424455987258612730378250981, −5.77650283157347906483553227232, −4.57804083638854168018764856816, −3.67970806585876338369986442533, −2.00105872890907254017453475198, −0.53801773283226405451169975630, 1.31405509403724276617299966708, 4.32991430296968072452642158496, 5.02530879379495981455347203768, 6.32612830405350093546503583558, 6.67017412616772564512357583035, 7.24244063815431364870750936503, 8.745879599651920820291761899893, 9.143751364661178763918398180318, 10.71327086190644627015808691326, 11.31998428637902213923064161367

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