Properties

Label 2-360-40.29-c1-0-22
Degree $2$
Conductor $360$
Sign $0.348 + 0.937i$
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  + (0.321 + 1.37i)2-s + (−1.79 + 0.884i)4-s + (−2.11 − 0.726i)5-s − 4.05i·7-s + (−1.79 − 2.18i)8-s + (0.321 − 3.14i)10-s − 0.985i·11-s − 4.94·13-s + (5.58 − 1.30i)14-s + (2.43 − 3.17i)16-s − 4.52i·17-s + 2.60i·19-s + (4.43 − 0.567i)20-s + (1.35 − 0.316i)22-s + 3.53i·23-s + ⋯
L(s)  = 1  + (0.227 + 0.973i)2-s + (−0.896 + 0.442i)4-s + (−0.945 − 0.324i)5-s − 1.53i·7-s + (−0.634 − 0.773i)8-s + (0.101 − 0.994i)10-s − 0.297i·11-s − 1.37·13-s + (1.49 − 0.348i)14-s + (0.608 − 0.793i)16-s − 1.09i·17-s + 0.597i·19-s + (0.991 − 0.126i)20-s + (0.289 − 0.0674i)22-s + 0.737i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.348 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515508 - 0.358186i\)
\(L(\frac12)\) \(\approx\) \(0.515508 - 0.358186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.321 - 1.37i)T \)
3 \( 1 \)
5 \( 1 + (2.11 + 0.726i)T \)
good7 \( 1 + 4.05iT - 7T^{2} \)
11 \( 1 + 0.985iT - 11T^{2} \)
13 \( 1 + 4.94T + 13T^{2} \)
17 \( 1 + 4.52iT - 17T^{2} \)
19 \( 1 - 2.60iT - 19T^{2} \)
23 \( 1 - 3.53iT - 23T^{2} \)
29 \( 1 + 7.59iT - 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 - 0.945T + 37T^{2} \)
41 \( 1 + 0.568T + 41T^{2} \)
43 \( 1 + 8.45T + 43T^{2} \)
47 \( 1 + 2.60iT - 47T^{2} \)
53 \( 1 - 0.229T + 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 - 8.45T + 67T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 + 11.9iT - 73T^{2} \)
79 \( 1 - 3.28T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−11.45136069876150635318286011323, −10.14136227678360720112744930046, −9.341216376434225342517312728364, −7.982589326297066977931987923622, −7.52802702286445053638182974961, −6.76723287610848661273650751553, −5.21524936965601010373350502873, −4.36347047387001100555464289813, −3.43687726664174114817933444810, −0.39259526158820347161299115304, 2.19010116685950845437714571762, 3.19112753406882920780044722043, 4.54964378579797929068876697320, 5.44105061759257269436399407651, 6.82189816448155959118084237339, 8.207865890739164026068846006429, 8.912812884343134730364818024993, 9.916598911018470403072561076241, 10.90048234517454770658564513788, 11.71613133451437851252561251168

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