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Properties

Label 2-360-5.3-c2-0-3
Degree $2$
Conductor $360$
Sign $0.130 - 0.991i$
Analytic cond. $9.80928$
Root an. cond. $3.13197$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.67 − 1.77i)5-s + (−0.550 + 0.550i)7-s + 1.55·11-s + (9.55 + 9.55i)13-s + (−11.1 + 11.1i)17-s + 12.6i·19-s + (21.3 + 21.3i)23-s + (18.6 + 16.5i)25-s + 44.0i·29-s − 44.4·31-s + (3.55 − 1.59i)35-s + (20.6 − 20.6i)37-s + 48.2·41-s + (−36.2 − 36.2i)43-s + (−42.5 + 42.5i)47-s + ⋯
L(s)  = 1  + (−0.934 − 0.355i)5-s + (−0.0786 + 0.0786i)7-s + 0.140·11-s + (0.734 + 0.734i)13-s + (−0.655 + 0.655i)17-s + 0.668i·19-s + (0.928 + 0.928i)23-s + (0.747 + 0.663i)25-s + 1.51i·29-s − 1.43·31-s + (0.101 − 0.0455i)35-s + (0.558 − 0.558i)37-s + 1.17·41-s + (−0.844 − 0.844i)43-s + (−0.905 + 0.905i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(9.80928\)
Root analytic conductor: \(3.13197\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.798225 + 0.699851i\)
\(L(\frac12)\) \(\approx\) \(0.798225 + 0.699851i\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (4.67 + 1.77i)T \)
good7 \( 1 + (0.550 - 0.550i)T - 49iT^{2} \)
11 \( 1 - 1.55T + 121T^{2} \)
13 \( 1 + (-9.55 - 9.55i)T + 169iT^{2} \)
17 \( 1 + (11.1 - 11.1i)T - 289iT^{2} \)
19 \( 1 - 12.6iT - 361T^{2} \)
23 \( 1 + (-21.3 - 21.3i)T + 529iT^{2} \)
29 \( 1 - 44.0iT - 841T^{2} \)
31 \( 1 + 44.4T + 961T^{2} \)
37 \( 1 + (-20.6 + 20.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 48.2T + 1.68e3T^{2} \)
43 \( 1 + (36.2 + 36.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (42.5 - 42.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (-54.4 - 54.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 47.4iT - 3.48e3T^{2} \)
61 \( 1 + 59.8T + 3.72e3T^{2} \)
67 \( 1 + (-81.2 + 81.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 87.5T + 5.04e3T^{2} \)
73 \( 1 + (-75.9 - 75.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 97.3iT - 6.24e3T^{2} \)
83 \( 1 + (41.0 + 41.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 52.2iT - 7.92e3T^{2} \)
97 \( 1 + (37 - 37i)T - 9.40e3iT^{2} \)
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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

−11.23463372785612039894977627677, −10.89553176565481940150302314017, −9.321113831998161912565250435256, −8.748491414926800431666763340718, −7.68748826888797374971924537916, −6.77160714442978600098162066185, −5.54453664242006789901159182228, −4.28771686646740682966287750492, −3.41277368703938139268315055614, −1.47602068081692117911327766613, 0.51468744433495763708339182356, 2.69047994464430628962362297522, 3.84004777194299258946568098510, 4.93804752988904653324311933070, 6.35084191558631068195004844095, 7.21939202735366245297563920405, 8.194645366701670809497984800963, 9.040084413986547971282894256909, 10.24779985025809962846982280298, 11.18191946075976649003083269411

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