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Properties

Label 2-1975-395.394-c0-0-3
Degree $2$
Conductor $1975$
Sign $0.447 + 0.894i$
Analytic cond. $0.985653$
Root an. cond. $0.992800$
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  + 1.61i·2-s − 1.61·4-s − i·8-s − 9-s − 1.61·11-s − 1.61i·13-s − 1.61i·18-s − 0.618·19-s − 2.61i·22-s + 0.618i·23-s + 2.61·26-s − 1.61·31-s − i·32-s + 1.61·36-s − 1.00i·38-s + ⋯
L(s)  = 1  + 1.61i·2-s − 1.61·4-s − i·8-s − 9-s − 1.61·11-s − 1.61i·13-s − 1.61i·18-s − 0.618·19-s − 2.61i·22-s + 0.618i·23-s + 2.61·26-s − 1.61·31-s − i·32-s + 1.61·36-s − 1.00i·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1975\)    =    \(5^{2} \cdot 79\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(0.985653\)
Root analytic conductor: \(0.992800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1975} (1974, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1975,\ (\ :0),\ 0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03697125282\)
\(L(\frac12)\) \(\approx\) \(0.03697125282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
79 \( 1 + T \)
good2 \( 1 - 1.61iT - T^{2} \)
3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 + 1.61iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 0.618T + T^{2} \)
23 \( 1 - 0.618iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.618iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618iT - T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
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

−8.835773701468435147153360461717, −8.064707500719258666781354775025, −7.84325823657376132945543632681, −6.94225707121889992903798731628, −5.83607964520597216631779331238, −5.51977287740885538396895729465, −4.87389026170591000067396495481, −3.46518184360324496114732800617, −2.47728253103133410443785772920, −0.02426107055162404012020213940, 1.86647075045885345822920846539, 2.53064405916649520857153098566, 3.42441867362929581086425294668, 4.41600050465062195330488982680, 5.15969279836730106289144859601, 6.20097218535694920580075245881, 7.25988627729544353865861835119, 8.323306009507282048502960736187, 8.943335239383761888326038294421, 9.608790585644413126444937228777

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