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Properties

Label 2-2575-515.514-c0-0-8
Degree $2$
Conductor $2575$
Sign $0.894 - 0.447i$
Analytic cond. $1.28509$
Root an. cond. $1.13361$
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 − 0.618i·7-s − i·8-s − 9-s − 1.61i·13-s + 1.00·14-s − 0.618i·17-s − 1.61i·18-s + 1.61·19-s − 1.61i·23-s + 2.61·26-s + 1.00i·28-s − 0.618·29-s − i·32-s + ⋯
L(s)  = 1  + 1.61i·2-s − 1.61·4-s − 0.618i·7-s − i·8-s − 9-s − 1.61i·13-s + 1.00·14-s − 0.618i·17-s − 1.61i·18-s + 1.61·19-s − 1.61i·23-s + 2.61·26-s + 1.00i·28-s − 0.618·29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2575\)    =    \(5^{2} \cdot 103\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(1.28509\)
Root analytic conductor: \(1.13361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2575} (2574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2575,\ (\ :0),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8610835248\)
\(L(\frac12)\) \(\approx\) \(0.8610835248\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
103 \( 1 - iT \)
good2 \( 1 - 1.61iT - T^{2} \)
3 \( 1 + T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.61iT - T^{2} \)
17 \( 1 + 0.618iT - T^{2} \)
19 \( 1 - 1.61T + T^{2} \)
23 \( 1 + 1.61iT - T^{2} \)
29 \( 1 + 0.618T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 0.618T + T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.61T + T^{2} \)
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

−8.769091110625717844128368260213, −8.155053898861688560750372342533, −7.58162541323742514453545539189, −6.91300816232583591562772877889, −6.06569836045488985398165529213, −5.34471136082406711280755671525, −4.89599302531962308949200619366, −3.60828616715751469062424193785, −2.70926795065180795614901110791, −0.58474809483828700665199015804, 1.45962501456639549771994323563, 2.22643944098281531365111010865, 3.26019411637008586421651795890, 3.79564373334265621820309216227, 4.96508385090288477066359231000, 5.62985045181703279365311278165, 6.66883669401795317600334993741, 7.69052579394784197687913513002, 8.710418809021807937058789306274, 9.253223074129126277524896025731

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