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Properties

Label 2-360-72.11-c1-0-23
Degree $2$
Conductor $360$
Sign $-0.0369 - 0.999i$
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  + (0.893 + 1.09i)2-s + (1.59 + 0.669i)3-s + (−0.401 + 1.95i)4-s + (−0.5 − 0.866i)5-s + (0.693 + 2.34i)6-s + (2.53 + 1.46i)7-s + (−2.50 + 1.31i)8-s + (2.10 + 2.13i)9-s + (0.502 − 1.32i)10-s + (−3.49 − 2.01i)11-s + (−1.95 + 2.86i)12-s + (0.913 − 0.527i)13-s + (0.661 + 4.08i)14-s + (−0.218 − 1.71i)15-s + (−3.67 − 1.57i)16-s − 4.29i·17-s + ⋯
L(s)  = 1  + (0.632 + 0.774i)2-s + (0.922 + 0.386i)3-s + (−0.200 + 0.979i)4-s + (−0.223 − 0.387i)5-s + (0.283 + 0.959i)6-s + (0.957 + 0.552i)7-s + (−0.886 + 0.463i)8-s + (0.701 + 0.713i)9-s + (0.158 − 0.418i)10-s + (−1.05 − 0.608i)11-s + (−0.564 + 0.825i)12-s + (0.253 − 0.146i)13-s + (0.176 + 1.09i)14-s + (−0.0564 − 0.443i)15-s + (−0.919 − 0.393i)16-s − 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0369 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0369 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0369 - 0.999i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.0369 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64551 + 1.70749i\)
\(L(\frac12)\) \(\approx\) \(1.64551 + 1.70749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.893 - 1.09i)T \)
3 \( 1 + (-1.59 - 0.669i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-2.53 - 1.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.49 + 2.01i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.913 + 0.527i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.29iT - 17T^{2} \)
19 \( 1 + 0.364T + 19T^{2} \)
23 \( 1 + (-2.16 - 3.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.13 + 5.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.12 - 2.95i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.135iT - 37T^{2} \)
41 \( 1 + (4.36 - 2.52i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.44 + 7.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.06 + 8.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (-2.61 + 1.51i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.96 + 2.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.31 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 0.247T + 73T^{2} \)
79 \( 1 + (-10.7 - 6.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (15.2 + 8.83i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 + (-4.44 + 7.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.79710398969003770730647491663, −10.89887972833000810053406552002, −9.476416116173560329389274811498, −8.531890972820500954158064000427, −8.065438353386815702818520889351, −7.15971160412883769901257024390, −5.44459427050401053549772103866, −4.93769783969726417641273754153, −3.64896124287130323864089493341, −2.47288578444304493909097094112, 1.56668821456793389094451448153, 2.74114599809497234191040465009, 3.96160659228247321756069405182, 4.88650732225768451560194317106, 6.41292045495196917793768667056, 7.52925520443022678721863433500, 8.361647892733697299202148483319, 9.510052424368733342377583117613, 10.63144043031774334455714982328, 10.99558225639723396152166898114

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