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Properties

Label 2-360-9.7-c1-0-11
Degree $2$
Conductor $360$
Sign $-0.682 + 0.730i$
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  + (−0.403 − 1.68i)3-s + (−0.5 − 0.866i)5-s + (0.596 − 1.03i)7-s + (−2.67 + 1.35i)9-s + (1.66 − 2.87i)11-s + (−0.853 − 1.47i)13-s + (−1.25 + 1.19i)15-s − 6.34·17-s + 1.32·19-s + (−1.98 − 0.588i)21-s + (−3.43 − 5.94i)23-s + (−0.499 + 0.866i)25-s + (3.36 + 3.95i)27-s + (−1.01 + 1.75i)29-s + (1.33 + 2.32i)31-s + ⋯
L(s)  = 1  + (−0.232 − 0.972i)3-s + (−0.223 − 0.387i)5-s + (0.225 − 0.390i)7-s + (−0.891 + 0.452i)9-s + (0.500 − 0.867i)11-s + (−0.236 − 0.410i)13-s + (−0.324 + 0.307i)15-s − 1.53·17-s + 0.303·19-s + (−0.432 − 0.128i)21-s + (−0.715 − 1.23i)23-s + (−0.0999 + 0.173i)25-s + (0.648 + 0.761i)27-s + (−0.188 + 0.326i)29-s + (0.240 + 0.416i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.682 + 0.730i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.682 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391016 - 0.900688i\)
\(L(\frac12)\) \(\approx\) \(0.391016 - 0.900688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.403 + 1.68i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-0.596 + 1.03i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.66 + 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.853 + 1.47i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.34T + 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (3.43 + 5.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.01 - 1.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.33 - 2.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 + (1.16 + 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.17 + 5.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.38 + 11.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.02T + 53T^{2} \)
59 \( 1 + (-5.83 - 10.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.86 + 8.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.28 - 9.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.06T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + (0.707 - 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.91 + 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
97 \( 1 + (8.12 - 14.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.23851084403297721759623113378, −10.44486604083270200859582519147, −8.888020669536530990209331114534, −8.345864727230848655095402064885, −7.24114426710579908336457456117, −6.40309837379931247392603756524, −5.31379447278667365148163704674, −4.01325598142236583907545408187, −2.36196969136163863145051512041, −0.68728305420562628566961483884, 2.33976794203741960542706778660, 3.89244387469709192370114444810, 4.68550884616174101006525334097, 5.91370454537385295608148231578, 6.94270860637556033208875516887, 8.157172911281384528678171132049, 9.345205702649213939392057819461, 9.752135009101130945931211860787, 11.05170329603814233374525708599, 11.50566778870359765574650546357

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