Properties

Degree $2$
Conductor $1575$
Sign $0.505 - 0.862i$
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 + (0.965 + 0.258i)3-s + (0.500 + 0.866i)6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.965 − 0.258i)13-s + (−0.866 + 0.500i)14-s + 1.00·16-s + (−0.965 + 0.258i)17-s + (0.258 + 0.965i)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 + (0.965 + 0.258i)3-s + (0.500 + 0.866i)6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.965 − 0.258i)13-s + (−0.866 + 0.500i)14-s + 1.00·16-s + (−0.965 + 0.258i)17-s + (0.258 + 0.965i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.505 - 0.862i$
Motivic weight: \(0\)
Character: $\chi_{1575} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.152836075\)
\(L(\frac12)\) \(\approx\) \(2.152836075\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.731734788295441394881879335742, −8.849960674761809066764065326974, −8.198667060408948049278215685875, −7.06951340148210184225591203193, −6.72660393028981075545833182367, −5.34966177541291652188423944316, −5.03308679443230626344713403363, −3.91792195142930063995557640940, −2.93760197554213190169260213367, −1.90488381638609017449048181607, 1.57445781972414155785619589187, 2.64313276421559857091952309918, 3.38515846241032667959857457831, 4.26547356717856522543424328296, 4.86092978831002374481985380777, 6.35848522524563005834287245575, 7.40619227602422334785438608118, 7.64746129874310785373661049847, 8.723318468672411640192532865994, 9.599944183183575691215041778483

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