Properties

Degree $2$
Conductor $1575$
Sign $-0.342 + 0.939i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)7-s − 9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.866 − 0.5i)13-s + (−0.499 − 0.866i)16-s i·17-s + (−0.5 − 0.866i)21-s + i·27-s − 0.999i·28-s + (1 + 1.73i)29-s + (0.866 − 0.5i)33-s + (−0.5 + 0.866i)36-s + ⋯
L(s)  = 1  i·3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)7-s − 9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.866 − 0.5i)13-s + (−0.499 − 0.866i)16-s i·17-s + (−0.5 − 0.866i)21-s + i·27-s − 0.999i·28-s + (1 + 1.73i)29-s + (0.866 − 0.5i)33-s + (−0.5 + 0.866i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.342 + 0.939i$
Motivic weight: \(0\)
Character: $\chi_{1575} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.281607311\)
\(L(\frac12)\) \(\approx\) \(1.281607311\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.419392947415911640140214923693, −8.480268154298305589164354944446, −7.36806660239463043594069774347, −7.19323677972223513016072040393, −6.30533529410754720474081293563, −5.19468295254110238710123648436, −4.72639573927937808472050199215, −2.98671574054719098737459256306, −1.98492320336306636557745181058, −1.07673924168719496661651400743, 2.05669390229901320017423026929, 3.01694497752613453450060376560, 4.02025632214192708359342399432, 4.67433901265314038003178145429, 5.79949499433153945011159035881, 6.51033319227279921106186720480, 7.74404023282028677716941777182, 8.374586620062652387799110053328, 8.897076713692946444073901278197, 9.861623537689002301336391450163

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