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Properties

Label 2-360-24.11-c1-0-15
Degree $2$
Conductor $360$
Sign $-0.355 + 0.934i$
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  + (1.06 − 0.933i)2-s + (0.257 − 1.98i)4-s − 5-s − 1.41i·7-s + (−1.57 − 2.34i)8-s + (−1.06 + 0.933i)10-s − 2.31i·11-s − 5.14i·13-s + (−1.32 − 1.50i)14-s + (−3.86 − 1.02i)16-s + 5.10i·17-s + 8.24·19-s + (−0.257 + 1.98i)20-s + (−2.16 − 2.46i)22-s − 0.969·23-s + ⋯
L(s)  = 1  + (0.751 − 0.660i)2-s + (0.128 − 0.991i)4-s − 0.447·5-s − 0.534i·7-s + (−0.557 − 0.830i)8-s + (−0.335 + 0.295i)10-s − 0.699i·11-s − 1.42i·13-s + (−0.352 − 0.401i)14-s + (−0.966 − 0.255i)16-s + 1.23i·17-s + 1.89·19-s + (−0.0575 + 0.443i)20-s + (−0.461 − 0.525i)22-s − 0.202·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.355 + 0.934i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.355 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00653 - 1.45999i\)
\(L(\frac12)\) \(\approx\) \(1.00653 - 1.45999i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.933i)T \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 + 2.31iT - 11T^{2} \)
13 \( 1 + 5.14iT - 13T^{2} \)
17 \( 1 - 5.10iT - 17T^{2} \)
19 \( 1 - 8.24T + 19T^{2} \)
23 \( 1 + 0.969T + 23T^{2} \)
29 \( 1 + 3.28T + 29T^{2} \)
31 \( 1 - 5.10iT - 31T^{2} \)
37 \( 1 - 3.69iT - 37T^{2} \)
41 \( 1 - 4.59iT - 41T^{2} \)
43 \( 1 - 3.21T + 43T^{2} \)
47 \( 1 - 9.52T + 47T^{2} \)
53 \( 1 + 7.21T + 53T^{2} \)
59 \( 1 + 0.862iT - 59T^{2} \)
61 \( 1 + 4.63iT - 61T^{2} \)
67 \( 1 - 5.28T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 8.01iT - 79T^{2} \)
83 \( 1 + 7.38iT - 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−11.13052364026066672376001526356, −10.53763918369042213510213152611, −9.640598309646521984382109929072, −8.317482776642585922251343620508, −7.33396261723203793007913804015, −6.00658638259091708753139456317, −5.16860538703249658573720200011, −3.80848591670168681926778503212, −3.05397306429230918862682647609, −1.04838367354528433070263598597, 2.43685981076376651903883206501, 3.83204474239191399918377400902, 4.85220506268491266369014046905, 5.81403786327861424284012717952, 7.15061946341287261750769043661, 7.51832584913898938130258655071, 8.953194889294652918714758119568, 9.555338075954533309955141975901, 11.31879094305004972971732762472, 11.83663981968142237879769376340

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