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Properties

Label 2-425-85.24-c2-0-50
Degree $2$
Conductor $425$
Sign $-0.522 + 0.852i$
Analytic cond. $11.5804$
Root an. cond. $3.40300$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.03 + 0.841i)2-s + (0.134 + 0.0897i)3-s + (0.590 + 0.590i)4-s + (0.197 + 0.295i)6-s + (−6.12 − 1.21i)7-s + (−2.66 − 6.42i)8-s + (−3.43 − 8.29i)9-s + (−12.1 + 8.11i)11-s + (0.0263 + 0.132i)12-s + (4.79 + 4.79i)13-s + (−11.4 − 7.62i)14-s − 18.6i·16-s + (−15.7 − 6.50i)17-s − 19.7i·18-s + (9.56 − 23.0i)19-s + ⋯
L(s)  = 1  + (1.01 + 0.420i)2-s + (0.0447 + 0.0299i)3-s + (0.147 + 0.147i)4-s + (0.0329 + 0.0492i)6-s + (−0.874 − 0.174i)7-s + (−0.332 − 0.803i)8-s + (−0.381 − 0.921i)9-s + (−1.10 + 0.737i)11-s + (0.00219 + 0.0110i)12-s + (0.369 + 0.369i)13-s + (−0.815 − 0.544i)14-s − 1.16i·16-s + (−0.923 − 0.382i)17-s − 1.09i·18-s + (0.503 − 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ -0.522 + 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.492497 - 0.879554i\)
\(L(\frac12)\) \(\approx\) \(0.492497 - 0.879554i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (15.7 + 6.50i)T \)
good2 \( 1 + (-2.03 - 0.841i)T + (2.82 + 2.82i)T^{2} \)
3 \( 1 + (-0.134 - 0.0897i)T + (3.44 + 8.31i)T^{2} \)
7 \( 1 + (6.12 + 1.21i)T + (45.2 + 18.7i)T^{2} \)
11 \( 1 + (12.1 - 8.11i)T + (46.3 - 111. i)T^{2} \)
13 \( 1 + (-4.79 - 4.79i)T + 169iT^{2} \)
19 \( 1 + (-9.56 + 23.0i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (10.8 - 7.27i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (32.3 - 6.44i)T + (776. - 321. i)T^{2} \)
31 \( 1 + (1.00 + 0.674i)T + (367. + 887. i)T^{2} \)
37 \( 1 + (-38.6 - 25.8i)T + (523. + 1.26e3i)T^{2} \)
41 \( 1 + (-6.13 + 30.8i)T + (-1.55e3 - 643. i)T^{2} \)
43 \( 1 + (-67.3 + 27.8i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (-10.4 - 10.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (4.77 + 1.97i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (26.1 - 10.8i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (81.0 + 16.1i)T + (3.43e3 + 1.42e3i)T^{2} \)
67 \( 1 - 44.5T + 4.48e3T^{2} \)
71 \( 1 + (-32.1 + 48.1i)T + (-1.92e3 - 4.65e3i)T^{2} \)
73 \( 1 + (-1.32 + 0.262i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (-24.7 + 16.5i)T + (2.38e3 - 5.76e3i)T^{2} \)
83 \( 1 + (-25.7 + 62.2i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (90.1 + 90.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (-13.1 - 66.3i)T + (-8.69e3 + 3.60e3i)T^{2} \)
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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

−10.72412454994337956298779047784, −9.490416652618584671329405879982, −9.185303283650714379292937146027, −7.49010423498484250691433799825, −6.68419942906061994191266637583, −5.86916072748216253831956965951, −4.81856554001312959936940382288, −3.81259759623651615810745207939, −2.72620834426648487284256126948, −0.28332990374468101582724456657, 2.33597386910193278950583245691, 3.21045420839820102088330805512, 4.30289914399119447954964143659, 5.60069476524126317300883828325, 6.00065772469750846088951287809, 7.77184659916199941348817361335, 8.374560114609519551535491677856, 9.561176087616348303180172677348, 10.80039267899170654372573926288, 11.18161343553669938133322716364

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