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Properties

Label 2-60e2-180.23-c0-0-1
Degree $2$
Conductor $3600$
Sign $0.746 - 0.665i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + (0.258 + 0.965i)3-s + (0.258 − 0.965i)7-s + (−0.866 + 0.499i)9-s + 21-s + (1.67 − 0.448i)23-s + (−0.707 − 0.707i)27-s + (0.866 + 1.5i)29-s + (1.5 + 0.866i)41-s + (−1.93 − 0.517i)43-s + (1.67 + 0.448i)47-s + (0.5 + 0.866i)61-s + (0.258 + 0.965i)63-s + (0.965 − 0.258i)67-s + (0.866 + 1.50i)69-s + (0.500 − 0.866i)81-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (0.258 − 0.965i)7-s + (−0.866 + 0.499i)9-s + 21-s + (1.67 − 0.448i)23-s + (−0.707 − 0.707i)27-s + (0.866 + 1.5i)29-s + (1.5 + 0.866i)41-s + (−1.93 − 0.517i)43-s + (1.67 + 0.448i)47-s + (0.5 + 0.866i)61-s + (0.258 + 0.965i)63-s + (0.965 − 0.258i)67-s + (0.866 + 1.50i)69-s + (0.500 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.443300816\)
\(L(\frac12)\) \(\approx\) \(1.443300816\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
good7 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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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

−8.843001075753144302337380564163, −8.245200107699561298374067054875, −7.32586620562079326157485376928, −6.73230625915502514091636166104, −5.61712528360994364551685714092, −4.84162502119233901552783854661, −4.29265145039428555527401189451, −3.38478544607371818413645973307, −2.64360837480741990100940618805, −1.13812322417201503930187416497, 1.03367402490081284037359720645, 2.23974975576718639749561806506, 2.79334281540399881339795367190, 3.88379050519461418042499571716, 5.09511535639075341659869383387, 5.68200726930464547364507863348, 6.51568175243000774570027600371, 7.15148578307249902086654288441, 7.986420971799502323774164008937, 8.547953908587324940413858068247

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