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Properties

Label 2-875-25.11-c1-0-1
Degree $2$
Conductor $875$
Sign $-0.538 - 0.842i$
Analytic cond. $6.98691$
Root an. cond. $2.64327$
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  + (−0.875 + 0.635i)2-s + (0.987 − 3.03i)3-s + (−0.256 + 0.789i)4-s + (1.06 + 3.28i)6-s − 7-s + (−0.945 − 2.91i)8-s + (−5.83 − 4.23i)9-s + (−3.82 + 2.78i)11-s + (2.14 + 1.55i)12-s + (1.22 + 0.888i)13-s + (0.875 − 0.635i)14-s + (1.33 + 0.970i)16-s + (0.193 + 0.595i)17-s + 7.80·18-s + (0.975 + 3.00i)19-s + ⋯
L(s)  = 1  + (−0.618 + 0.449i)2-s + (0.570 − 1.75i)3-s + (−0.128 + 0.394i)4-s + (0.436 + 1.34i)6-s − 0.377·7-s + (−0.334 − 1.02i)8-s + (−1.94 − 1.41i)9-s + (−1.15 + 0.838i)11-s + (0.619 + 0.450i)12-s + (0.339 + 0.246i)13-s + (0.233 − 0.169i)14-s + (0.333 + 0.242i)16-s + (0.0469 + 0.144i)17-s + 1.83·18-s + (0.223 + 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $-0.538 - 0.842i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ -0.538 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108267 + 0.197711i\)
\(L(\frac12)\) \(\approx\) \(0.108267 + 0.197711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + (0.875 - 0.635i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.987 + 3.03i)T + (-2.42 - 1.76i)T^{2} \)
11 \( 1 + (3.82 - 2.78i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.22 - 0.888i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.193 - 0.595i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.975 - 3.00i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.97 - 3.61i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.68 - 5.18i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.975 + 3.00i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.676 + 0.491i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.59 + 2.61i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.43T + 43T^{2} \)
47 \( 1 + (3.25 - 10.0i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.33 + 10.2i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.92 - 1.40i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.88 - 3.55i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.60 + 11.1i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.968 - 2.98i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.56 - 6.22i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.31 - 7.12i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.58 - 7.95i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (10.3 - 7.53i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.82 + 11.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.11103508101117080375725260326, −9.295227274782978342024426963860, −8.427134627825234941132387024594, −7.77214465307871781500742668729, −7.33649990356325018460936600825, −6.53339881958219950040315838123, −5.61858187589510067824818401496, −3.81010727789939646872292348178, −2.75920806243728386232803178200, −1.58812234143952625038431901529, 0.11884209655217425478604111544, 2.46262946531408766401362715902, 3.23223802304357000833438619780, 4.41327722858845626988622013910, 5.31919074277574943233243301506, 6.00251038651190546643849502964, 7.85316773582567183816268148400, 8.618765747751467584072297165862, 9.108650543258525848675893243855, 10.08989431957348875988488499353

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