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Properties

Label 2-875-7.6-c0-0-1
Degree $2$
Conductor $875$
Sign $1$
Analytic cond. $0.436681$
Root an. cond. $0.660819$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618i·3-s − 4-s + i·7-s + 0.618·9-s + 0.618·11-s + 0.618i·12-s + 1.61i·13-s + 16-s − 1.61i·17-s + 0.618·21-s − i·27-s − i·28-s + 1.61·29-s − 0.381i·33-s − 0.618·36-s + ⋯
L(s)  = 1  − 0.618i·3-s − 4-s + i·7-s + 0.618·9-s + 0.618·11-s + 0.618i·12-s + 1.61i·13-s + 16-s − 1.61i·17-s + 0.618·21-s − i·27-s − i·28-s + 1.61·29-s − 0.381i·33-s − 0.618·36-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.436681\)
Root analytic conductor: \(0.660819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8745300170\)
\(L(\frac12)\) \(\approx\) \(0.8745300170\)
\(L(1)\) 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 \( 1 - iT \)
good2 \( 1 + T^{2} \)
3 \( 1 + 0.618iT - T^{2} \)
11 \( 1 - 0.618T + T^{2} \)
13 \( 1 - 1.61iT - T^{2} \)
17 \( 1 + 1.61iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.61T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 + 0.618T + T^{2} \)
83 \( 1 + 0.618iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 0.618iT - T^{2} \)
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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

−10.06313991300824373701197233344, −9.271614585723471420269510940290, −8.927043313904680938648300310019, −7.84593106414548627567324279982, −6.88893723076757635586790227487, −6.13588825924972579016796755000, −4.86850242508771239631128537194, −4.28491666971772843521008193689, −2.78856612472768794868915354127, −1.41170299502429541264085753605, 1.16340452833181946038778872685, 3.39548853972808677237144362086, 4.04156429358013025713332891011, 4.81938247619544336741122818402, 5.84457927608184862165346145595, 6.98621280083361313773473043907, 8.055710758659533259737499954967, 8.617307391726833924281717386220, 9.758430093982436002360197829663, 10.33694630513849921390906346374

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