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Properties

Label 2-360-40.27-c1-0-20
Degree $2$
Conductor $360$
Sign $0.376 + 0.926i$
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  + (0.909 − 1.08i)2-s + (−0.345 − 1.96i)4-s + (0.780 + 2.09i)5-s + (2.10 − 2.10i)7-s + (−2.44 − 1.41i)8-s + (2.97 + 1.05i)10-s + 3.11·11-s + (2.19 + 2.19i)13-s + (−0.366 − 4.20i)14-s + (−3.76 + 1.36i)16-s + (−5.48 − 5.48i)17-s − 3.91i·19-s + (3.85 − 2.26i)20-s + (2.83 − 3.37i)22-s + (2.42 + 2.42i)23-s + ⋯
L(s)  = 1  + (0.643 − 0.765i)2-s + (−0.172 − 0.984i)4-s + (0.349 + 0.937i)5-s + (0.796 − 0.796i)7-s + (−0.865 − 0.500i)8-s + (0.942 + 0.335i)10-s + 0.940·11-s + (0.608 + 0.608i)13-s + (−0.0978 − 1.12i)14-s + (−0.940 + 0.340i)16-s + (−1.33 − 1.33i)17-s − 0.899i·19-s + (0.862 − 0.505i)20-s + (0.604 − 0.720i)22-s + (0.505 + 0.505i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69973 - 1.14442i\)
\(L(\frac12)\) \(\approx\) \(1.69973 - 1.14442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 + 1.08i)T \)
3 \( 1 \)
5 \( 1 + (-0.780 - 2.09i)T \)
good7 \( 1 + (-2.10 + 2.10i)T - 7iT^{2} \)
11 \( 1 - 3.11T + 11T^{2} \)
13 \( 1 + (-2.19 - 2.19i)T + 13iT^{2} \)
17 \( 1 + (5.48 + 5.48i)T + 17iT^{2} \)
19 \( 1 + 3.91iT - 19T^{2} \)
23 \( 1 + (-2.42 - 2.42i)T + 23iT^{2} \)
29 \( 1 - 2.23T + 29T^{2} \)
31 \( 1 - 7.17iT - 31T^{2} \)
37 \( 1 + (1.23 - 1.23i)T - 37iT^{2} \)
41 \( 1 + 4.40T + 41T^{2} \)
43 \( 1 + (1.50 - 1.50i)T - 43iT^{2} \)
47 \( 1 + (2.00 - 2.00i)T - 47iT^{2} \)
53 \( 1 + (5.56 + 5.56i)T + 53iT^{2} \)
59 \( 1 - 5.44iT - 59T^{2} \)
61 \( 1 - 7.46iT - 61T^{2} \)
67 \( 1 + (-6.40 - 6.40i)T + 67iT^{2} \)
71 \( 1 + 2.00iT - 71T^{2} \)
73 \( 1 + (1.49 - 1.49i)T - 73iT^{2} \)
79 \( 1 + 1.91T + 79T^{2} \)
83 \( 1 + (-6.67 + 6.67i)T - 83iT^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + (10.0 + 10.0i)T + 97iT^{2} \)
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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

−11.25384649532824684080708716012, −10.74613264683603170717643233161, −9.614701526182944123978300642460, −8.808880194656674368580813777078, −6.99878783293117322283949931114, −6.60765595217286659210486816618, −5.05688375445247998811302420954, −4.13920798208262122842163452157, −2.90820175634852960897456504118, −1.49288824239950048792504829538, 1.93622529933491814958763325552, 3.83875245850982371519118673986, 4.79752492907550687350917843912, 5.80799292612559102652345614297, 6.49534073185806018115404888919, 8.178729126223456258603656385920, 8.472314428651096821223450055578, 9.392912295084123618965785569250, 10.94045190767900349494785107882, 11.91169040938519236534500800923

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