Properties

Label 2-360-45.4-c1-0-5
Degree $2$
Conductor $360$
Sign $0.937 - 0.347i$
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  + (−1.72 + 0.131i)3-s + (2.23 − 0.0643i)5-s + (−1.19 + 0.687i)7-s + (2.96 − 0.454i)9-s + (−0.00579 − 0.0100i)11-s + (0.919 + 0.530i)13-s + (−3.85 + 0.405i)15-s + 1.57i·17-s + 6.38·19-s + (1.96 − 1.34i)21-s + (6.37 + 3.68i)23-s + (4.99 − 0.287i)25-s + (−5.06 + 1.17i)27-s + (2.66 + 4.61i)29-s + (1.15 − 2.00i)31-s + ⋯
L(s)  = 1  + (−0.997 + 0.0759i)3-s + (0.999 − 0.0287i)5-s + (−0.450 + 0.259i)7-s + (0.988 − 0.151i)9-s + (−0.00174 − 0.00302i)11-s + (0.254 + 0.147i)13-s + (−0.994 + 0.104i)15-s + 0.382i·17-s + 1.46·19-s + (0.429 − 0.293i)21-s + (1.32 + 0.767i)23-s + (0.998 − 0.0574i)25-s + (−0.974 + 0.226i)27-s + (0.494 + 0.856i)29-s + (0.207 − 0.359i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.937 - 0.347i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15133 + 0.206784i\)
\(L(\frac12)\) \(\approx\) \(1.15133 + 0.206784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.131i)T \)
5 \( 1 + (-2.23 + 0.0643i)T \)
good7 \( 1 + (1.19 - 0.687i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.00579 + 0.0100i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.919 - 0.530i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.57iT - 17T^{2} \)
19 \( 1 - 6.38T + 19T^{2} \)
23 \( 1 + (-6.37 - 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.66 - 4.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.15 + 2.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + (4.09 - 7.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.93 + 1.69i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.60 - 2.66i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.27iT - 53T^{2} \)
59 \( 1 + (-3.39 + 5.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.29 + 7.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.16 + 4.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 16.4T + 71T^{2} \)
73 \( 1 + 8.32iT - 73T^{2} \)
79 \( 1 + (-3.86 - 6.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.67 + 2.69i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (0.0741 - 0.0428i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.39072498781796609740747455621, −10.63419652763719416252021291345, −9.657487100067980117753483818323, −9.104500599612459863535928442578, −7.46656410852885657350347981093, −6.48757856422428473555981910796, −5.68029060595247216558823971710, −4.86193918003165440961755151113, −3.20441283430644385273875638117, −1.38614608496536308978668374148, 1.14188732555710000839412851748, 2.98520669466063236847049372074, 4.70452366286656992101055643180, 5.57227011057042742740758841033, 6.53019862101364649007896451246, 7.25417326173495252753122747980, 8.774503504301377073876381028518, 9.897183060363345668397112161432, 10.31332736520112334278041755733, 11.41372219666366195910897872524

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