Properties

Degree $2$
Conductor $1575$
Sign $-0.837 + 0.545i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.258 − 0.965i)3-s + (0.500 + 0.866i)6-s + (−0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.258 + 0.965i)13-s + (0.866 − 0.500i)14-s + 1.00·16-s + (−0.258 − 0.965i)17-s + (0.965 − 0.258i)18-s + (−0.866 + 0.5i)19-s + (−0.499 + 0.866i)21-s + (−0.866 + 0.5i)24-s + (−0.500 − 0.866i)26-s + (−0.707 + 0.707i)27-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.258 − 0.965i)3-s + (0.500 + 0.866i)6-s + (−0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.258 + 0.965i)13-s + (0.866 − 0.500i)14-s + 1.00·16-s + (−0.258 − 0.965i)17-s + (0.965 − 0.258i)18-s + (−0.866 + 0.5i)19-s + (−0.499 + 0.866i)21-s + (−0.866 + 0.5i)24-s + (−0.500 − 0.866i)26-s + (−0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.837 + 0.545i$
Motivic weight: \(0\)
Character: $\chi_{1575} (718, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.837 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1276778588\)
\(L(\frac12)\) \(\approx\) \(0.1276778588\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.965 + 0.258i)T \)
good2 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.077015504427993223130327199205, −8.529684826902436470747356274013, −7.45437096590041251792979008151, −7.08527550923236829500419893813, −6.45662592285328315566848526210, −5.63494536132755835831409580040, −4.03797993435311022904855569012, −3.15398381557730017306994834060, −1.96924782360785571842396142702, −0.11204262707789757538186926695, 1.98491876706737872351358226189, 2.99856500217957436810707241741, 3.70719740993966932509469020224, 5.02626659687288173713339754663, 5.75021382421293909511570647798, 6.62820299876220975412032235697, 8.012424276037981049955787168389, 8.687857742345498212085428496068, 9.328136589317893312844647121106, 9.906057537220594783757527754972

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