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Properties

Label 2-1575-21.17-c1-0-35
Degree $2$
Conductor $1575$
Sign $0.879 + 0.475i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)4-s + (2 + 1.73i)7-s + (−3.67 − 2.12i)11-s − 3.46i·13-s + (−1.99 − 3.46i)16-s + (3.67 − 6.36i)17-s + (6 − 3.46i)19-s + (−7.34 + 4.24i)23-s + (−5 + 1.73i)28-s − 8.48i·29-s + (4.5 + 2.59i)31-s + (−0.5 − 0.866i)37-s + 7.34·41-s + 43-s + (7.34 − 4.24i)44-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (0.755 + 0.654i)7-s + (−1.10 − 0.639i)11-s − 0.960i·13-s + (−0.499 − 0.866i)16-s + (0.891 − 1.54i)17-s + (1.37 − 0.794i)19-s + (−1.53 + 0.884i)23-s + (−0.944 + 0.327i)28-s − 1.57i·29-s + (0.808 + 0.466i)31-s + (−0.0821 − 0.142i)37-s + 1.14·41-s + 0.152·43-s + (1.10 − 0.639i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.393568961\)
\(L(\frac12)\) \(\approx\) \(1.393568961\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
11 \( 1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-3.67 + 6.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.34 - 4.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-3.67 - 6.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.67 - 2.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 + 6.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.24iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 + (3.67 + 6.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−9.397850517786304520159872625695, −8.353661754441050316955358157498, −7.78315580079026032554076455383, −7.44859732480358217510038179436, −5.72553459627100077926319626327, −5.33835117642765699830506416454, −4.39891108947549744041239091757, −3.08450182223212347988508345429, −2.63120028386551883008523547103, −0.63366708554518095125514263815, 1.19042539800226653739321802644, 2.09467888974689179794763246238, 3.79482389896440824575017255812, 4.49660248596096799821417721660, 5.37393017881720374457616197874, 6.04870581732848160975976016413, 7.18541008724191492390594875736, 7.955649501818552763499490855784, 8.606535310969495157261145930224, 9.730538205933731428885493250993

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