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Properties

Label 2-1575-315.277-c0-0-0
Degree $2$
Conductor $1575$
Sign $-0.505 + 0.862i$
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  + (−0.707 − 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: $-0.505 + 0.862i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.505 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5982760658\)
\(L(\frac12)\) \(\approx\) \(0.5982760658\)
\(L(1)\) 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

−9.603319138827349392277281208194, −8.744087307891417081145011220106, −7.70022553231134499543271460783, −7.07438901958422040681689379389, −6.02071068137581853284546568107, −5.35284685739647334381662094454, −4.33607874120362633048353714018, −3.16003554654582166684090138636, −1.62785111165918555781214206912, −0.855545169668131933687633899687, 1.25884491901131651590652254621, 3.09969212426919849260878773271, 4.05167961364493317466960213993, 5.42284426143356360275084181331, 5.80431984061564972039347097281, 6.64909357074799859776883621271, 7.52511449678132289602629026028, 8.327132400615418582722418772737, 8.940947355638329889183173267826, 9.875493879717417614409662989809

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