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Properties

Label 2-875-25.21-c1-0-24
Degree $2$
Conductor $875$
Sign $0.899 + 0.436i$
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  + (0.591 − 1.82i)2-s + (2.32 + 1.68i)3-s + (−1.34 − 0.980i)4-s + (4.45 − 3.23i)6-s − 7-s + (0.513 − 0.373i)8-s + (1.62 + 4.99i)9-s + (−0.946 + 2.91i)11-s + (−1.48 − 4.55i)12-s + (1.86 + 5.74i)13-s + (−0.591 + 1.82i)14-s + (−1.40 − 4.33i)16-s + (1.11 − 0.811i)17-s + 10.0·18-s + (5.21 − 3.79i)19-s + ⋯
L(s)  = 1  + (0.418 − 1.28i)2-s + (1.34 + 0.975i)3-s + (−0.674 − 0.490i)4-s + (1.81 − 1.32i)6-s − 0.377·7-s + (0.181 − 0.131i)8-s + (0.541 + 1.66i)9-s + (−0.285 + 0.877i)11-s + (−0.427 − 1.31i)12-s + (0.517 + 1.59i)13-s + (−0.158 + 0.486i)14-s + (−0.351 − 1.08i)16-s + (0.270 − 0.196i)17-s + 2.37·18-s + (1.19 − 0.869i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ 0.899 + 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.02959 - 0.695653i\)
\(L(\frac12)\) \(\approx\) \(3.02959 - 0.695653i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.591 + 1.82i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-2.32 - 1.68i)T + (0.927 + 2.85i)T^{2} \)
11 \( 1 + (0.946 - 2.91i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.86 - 5.74i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.11 + 0.811i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-5.21 + 3.79i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.613 + 1.88i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.48 + 1.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.823 - 0.597i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.09 + 9.53i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.44 - 7.52i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.42T + 43T^{2} \)
47 \( 1 + (1.46 + 1.06i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.89 + 5.00i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.23 + 3.81i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.853 - 2.62i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (6.22 - 4.52i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-7.00 - 5.08i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.06 + 15.6i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.563 + 0.409i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.22 - 3.79i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.35 - 4.17i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (7.94 + 5.77i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.915346907950550863262627316953, −9.506943698524848247122795306218, −8.926546743256573645651750033997, −7.68511055192626772008003380332, −6.82983536740099196791959058356, −5.02140554451009832513042722960, −4.30426595981134175720027952786, −3.53920550343441067116515447742, −2.71082676053579283295121671612, −1.78794363507794632743021384820, 1.35953183927259612879304254097, 3.00221333614031743665165540657, 3.59466195625025635391806606662, 5.34225910448768792384267235925, 5.97224318792479450143546823871, 6.91928406854335143958086872977, 7.81004445446623817657301094948, 8.064859567296368007949353855403, 8.851871530975244351358814883907, 9.957578143304412801887266059582

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