↑

Properties

Label 2-1575-105.62-c0-0-7
Degree $2$
Conductor $1575$
Sign $0.583 + 0.812i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 1.36i)2-s − 2.73i·4-s + (−0.707 − 0.707i)7-s + (2.36 + 2.36i)8-s − 0.517i·11-s + 1.93·14-s − 3.73·16-s + (0.707 + 0.707i)22-s + (−0.366 − 0.366i)23-s + (−1.93 + 1.93i)28-s − 1.93·29-s + (2.73 − 2.73i)32-s + (−0.707 − 0.707i)37-s + (0.707 − 0.707i)43-s − 1.41·44-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s − 2.73i·4-s + (−0.707 − 0.707i)7-s + (2.36 + 2.36i)8-s − 0.517i·11-s + 1.93·14-s − 3.73·16-s + (0.707 + 0.707i)22-s + (−0.366 − 0.366i)23-s + (−1.93 + 1.93i)28-s − 1.93·29-s + (2.73 − 2.73i)32-s + (−0.707 − 0.707i)37-s + (0.707 − 0.707i)43-s − 1.41·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.583 + 0.812i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.583 + 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2852330630\)
\(L(\frac12)\) \(\approx\) \(0.2852330630\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
11 \( 1 + 0.517iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
29 \( 1 + 1.93T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
71 \( 1 + 1.93iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.258210552686842361387966865677, −8.768560239250328309365549497311, −7.69465797282092688826871085600, −7.35828359609446009750833086131, −6.38553524619062238732910335321, −5.88884031041120793783999873834, −4.87879346366663723314707384217, −3.60518447905527237955920397297, −1.85569189072496565439326249440, −0.35181967421435186547135899248, 1.55719241926109969703529944622, 2.51670352312665363395238879815, 3.36487174547964856459707515691, 4.28736615428135818115492047015, 5.71125091642555351704917283377, 6.92522307510639608588181929187, 7.62937510917383918796076045754, 8.484422849612557562650996831035, 9.218752980130347223413611626749, 9.680638297385790888210093623548

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