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Properties

Label 2-1575-315.67-c0-0-1
Degree $2$
Conductor $1575$
Sign $0.762 + 0.647i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (0.707 − 0.707i)3-s + (0.500 − 0.866i)6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (−0.258 − 0.965i)13-s + (0.866 + 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.258 − 0.965i)18-s + (0.866 − 0.5i)19-s + 1.00·21-s + 0.999i·24-s + (−0.499 − 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.707 − 0.707i)3-s + (0.500 − 0.866i)6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (−0.258 − 0.965i)13-s + (0.866 + 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.258 − 0.965i)18-s + (0.866 − 0.5i)19-s + 1.00·21-s + 0.999i·24-s + (−0.499 − 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.762 + 0.647i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (382, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.762 + 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.144694092\)
\(L(\frac12)\) \(\approx\) \(2.144694092\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.274670417507325236327233620610, −8.700647397858995282517722512812, −7.893323417589974983944957441709, −7.30517919104305582803904121734, −5.97872240584779656472603646679, −5.38155020608133063839802400433, −4.53028499009629204855373191096, −3.18708264890501264989660077222, −2.88977262446882646130614433621, −1.54377635240555344170191146008, 1.72536222333760870651217469146, 3.23924089804624685617367967777, 3.88627037559105298383878259479, 4.68094067830853605853580437579, 5.26036485602424613262808798955, 6.29589172712385127514043180874, 7.38804618673787384776763766208, 8.020520536251053597589218834887, 8.972536880883109640724116207594, 9.864101953592230753143292206540

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