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Properties

Label 2-1975-395.394-c0-0-1
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  + 0.618i·2-s + 0.618·4-s + i·8-s − 9-s + 0.618·11-s − 0.618i·13-s − 0.618i·18-s + 1.61·19-s + 0.381i·22-s + 1.61i·23-s + 0.381·26-s + 0.618·31-s + 0.999i·32-s − 0.618·36-s + 1.00i·38-s + ⋯
L(s)  = 1  + 0.618i·2-s + 0.618·4-s + i·8-s − 9-s + 0.618·11-s − 0.618i·13-s − 0.618i·18-s + 1.61·19-s + 0.381i·22-s + 1.61i·23-s + 0.381·26-s + 0.618·31-s + 0.999i·32-s − 0.618·36-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

\(L(\frac{1}{2})\) \(\approx\) \(1.388118062\)
\(L(\frac12)\) \(\approx\) \(1.388118062\)
\(L(1)\) 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 \( 1 - 0.618iT - T^{2} \)
3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 0.618T + T^{2} \)
13 \( 1 + 0.618iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.61T + T^{2} \)
23 \( 1 - 1.61iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
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 \( 1 + 1.61iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.61iT - T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 + 1.61iT - T^{2} \)
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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

−9.403982628873421685553483037303, −8.550738355413035070585920306535, −7.78385315173531602656301683909, −7.24970083687714733744835500368, −6.26670983509683467230463290888, −5.65148346643809418653664292336, −5.00512660658087552940120361320, −3.47830670147833745669285728142, −2.84189087946226129861056641964, −1.47711377691477051913384929197, 1.15282602885632469257823728687, 2.41038003220776168509411392583, 3.13526401956997833404193791357, 4.08831533484336784572638961913, 5.18687773721832427217101701105, 6.21390436680707159320100339821, 6.73784434404710056966631771293, 7.66115581542961393213803750318, 8.558828613359730197367444142451, 9.342316748409619706469807320672

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