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Properties

Label 2-875-175.117-c1-0-25
Degree $2$
Conductor $875$
Sign $0.956 + 0.292i$
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  + (0.504 + 0.776i)2-s + (−0.473 − 1.23i)3-s + (0.464 − 1.04i)4-s + (0.718 − 0.989i)6-s + (0.0993 + 2.64i)7-s + (2.87 − 0.455i)8-s + (0.932 − 0.839i)9-s + (−0.526 + 0.584i)11-s + (−1.50 − 0.0790i)12-s + (0.729 − 0.371i)13-s + (−2.00 + 1.41i)14-s + (0.272 + 0.303i)16-s + (4.67 + 5.77i)17-s + (1.12 + 0.300i)18-s + (1.68 − 0.748i)19-s + ⋯
L(s)  = 1  + (0.356 + 0.549i)2-s + (−0.273 − 0.712i)3-s + (0.232 − 0.522i)4-s + (0.293 − 0.403i)6-s + (0.0375 + 0.999i)7-s + (1.01 − 0.160i)8-s + (0.310 − 0.279i)9-s + (−0.158 + 0.176i)11-s + (−0.435 − 0.0228i)12-s + (0.202 − 0.103i)13-s + (−0.535 + 0.376i)14-s + (0.0682 + 0.0757i)16-s + (1.13 + 1.40i)17-s + (0.264 + 0.0708i)18-s + (0.385 − 0.171i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ 0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07110 - 0.309672i\)
\(L(\frac12)\) \(\approx\) \(2.07110 - 0.309672i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0993 - 2.64i)T \)
good2 \( 1 + (-0.504 - 0.776i)T + (-0.813 + 1.82i)T^{2} \)
3 \( 1 + (0.473 + 1.23i)T + (-2.22 + 2.00i)T^{2} \)
11 \( 1 + (0.526 - 0.584i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.729 + 0.371i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-4.67 - 5.77i)T + (-3.53 + 16.6i)T^{2} \)
19 \( 1 + (-1.68 + 0.748i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-3.42 + 2.22i)T + (9.35 - 21.0i)T^{2} \)
29 \( 1 + (5.50 + 7.57i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.69 - 0.493i)T + (30.3 + 6.44i)T^{2} \)
37 \( 1 + (-0.343 + 6.55i)T + (-36.7 - 3.86i)T^{2} \)
41 \( 1 + (2.40 + 0.783i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.79 - 1.79i)T - 43iT^{2} \)
47 \( 1 + (-8.95 - 7.25i)T + (9.77 + 45.9i)T^{2} \)
53 \( 1 + (8.40 - 3.22i)T + (39.3 - 35.4i)T^{2} \)
59 \( 1 + (-11.1 + 2.36i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-1.75 + 8.26i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (-0.709 + 0.574i)T + (13.9 - 65.5i)T^{2} \)
71 \( 1 + (-0.767 + 0.557i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.70 - 0.194i)T + (72.6 - 7.63i)T^{2} \)
79 \( 1 + (4.86 - 0.511i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (-0.814 - 5.14i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (0.565 + 0.120i)T + (81.3 + 36.1i)T^{2} \)
97 \( 1 + (1.67 - 10.5i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.07559467912403144730479422029, −9.322677557710249895216961245771, −8.095509479065637369782251602982, −7.46114659054655610548451539463, −6.39170567570814820408596069315, −5.95360782377007672997685576289, −5.14698446584817776855682010243, −3.89207129214313297259476653515, −2.29708450769552356890500362453, −1.17291632200532060081642448181, 1.36105105690743739086199748614, 3.02893558588132704055386909985, 3.73651334207676833111138489938, 4.71761531535910240609643372219, 5.39456161856476509978545991005, 7.12584149262134242142443258259, 7.37458744556272550845934187751, 8.503533416748704241329343660156, 9.777428722261974219909676582347, 10.23686324646142742739703208774

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