Properties

Label 2-360-120.77-c1-0-19
Degree $2$
Conductor $360$
Sign $0.878 + 0.478i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 0.159i)2-s + (1.94 + 0.449i)4-s + (0.215 − 2.22i)5-s + (−2.32 − 2.32i)7-s + (2.66 + 0.943i)8-s + (0.658 − 3.09i)10-s + 5.57·11-s + (−1.79 − 1.79i)13-s + (−2.88 − 3.63i)14-s + (3.59 + 1.75i)16-s + (−5.56 + 5.56i)17-s + 2.61·19-s + (1.42 − 4.24i)20-s + (7.83 + 0.892i)22-s + (4.44 + 4.44i)23-s + ⋯
L(s)  = 1  + (0.993 + 0.113i)2-s + (0.974 + 0.224i)4-s + (0.0963 − 0.995i)5-s + (−0.877 − 0.877i)7-s + (0.942 + 0.333i)8-s + (0.208 − 0.978i)10-s + 1.68·11-s + (−0.497 − 0.497i)13-s + (−0.772 − 0.970i)14-s + (0.898 + 0.438i)16-s + (−1.34 + 1.34i)17-s + 0.600·19-s + (0.317 − 0.948i)20-s + (1.67 + 0.190i)22-s + (0.927 + 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.878 + 0.478i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.878 + 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31927 - 0.590306i\)
\(L(\frac12)\) \(\approx\) \(2.31927 - 0.590306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.159i)T \)
3 \( 1 \)
5 \( 1 + (-0.215 + 2.22i)T \)
good7 \( 1 + (2.32 + 2.32i)T + 7iT^{2} \)
11 \( 1 - 5.57T + 11T^{2} \)
13 \( 1 + (1.79 + 1.79i)T + 13iT^{2} \)
17 \( 1 + (5.56 - 5.56i)T - 17iT^{2} \)
19 \( 1 - 2.61T + 19T^{2} \)
23 \( 1 + (-4.44 - 4.44i)T + 23iT^{2} \)
29 \( 1 - 2.12iT - 29T^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 + (5.02 - 5.02i)T - 37iT^{2} \)
41 \( 1 + 1.73iT - 41T^{2} \)
43 \( 1 + (1.19 + 1.19i)T + 43iT^{2} \)
47 \( 1 + (-0.849 + 0.849i)T - 47iT^{2} \)
53 \( 1 + (-4.22 + 4.22i)T - 53iT^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 + 3.13iT - 61T^{2} \)
67 \( 1 + (-1.86 + 1.86i)T - 67iT^{2} \)
71 \( 1 + 6.95iT - 71T^{2} \)
73 \( 1 + (5.86 - 5.86i)T - 73iT^{2} \)
79 \( 1 - 2.52iT - 79T^{2} \)
83 \( 1 + (-0.694 + 0.694i)T - 83iT^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-4.09 - 4.09i)T + 97iT^{2} \)
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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

−11.63479876261404089283992904337, −10.60809794000123119730307926095, −9.538956359954253917108925957860, −8.593514441127183796868864223902, −7.21357304268945216901459288025, −6.52842843313867358489418722068, −5.40903870999935819509803168301, −4.22482008508185271797410959346, −3.50968601791068556682907873690, −1.50740168261011656596091939244, 2.27305620657179080417394191866, 3.21472854375194617667845707323, 4.41586726009977277392367330557, 5.76819138923684142197708134372, 6.80420153689834993362923630366, 7.00410620393266835639326392099, 9.052824735242427777538854759356, 9.651289913200651366055014551200, 10.96025393712127106233092046617, 11.64644209942577896599345459661

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