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Properties

Label 2-360-360.317-c1-0-63
Degree $2$
Conductor $360$
Sign $-0.0572 - 0.998i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−0.866 − 1.5i)3-s + 2i·4-s + (1.23 − 1.86i)5-s + (−0.633 + 2.36i)6-s + (−4.23 + 1.13i)7-s + (2 − 2i)8-s + (−1.5 + 2.59i)9-s + (−3.09 + 0.633i)10-s + (−1 + 1.73i)11-s + (3 − 1.73i)12-s + (0.464 − 1.73i)13-s + (5.36 + 3.09i)14-s + (−3.86 − 0.232i)15-s − 4·16-s + (−3 + 3i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.499 − 0.866i)3-s + i·4-s + (0.550 − 0.834i)5-s + (−0.258 + 0.965i)6-s + (−1.59 + 0.428i)7-s + (0.707 − 0.707i)8-s + (−0.5 + 0.866i)9-s + (−0.979 + 0.200i)10-s + (−0.301 + 0.522i)11-s + (0.866 − 0.499i)12-s + (0.128 − 0.480i)13-s + (1.43 + 0.827i)14-s + (−0.998 − 0.0599i)15-s − 16-s + (−0.727 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0572 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0572 - 0.998i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.0572 - 0.998i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + (-1.23 + 1.86i)T \)
good7 \( 1 + (4.23 - 1.13i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.464 + 1.73i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 + (-0.232 + 0.866i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.76 - 1.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.09 - 7.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.464 - 0.464i)T - 37iT^{2} \)
41 \( 1 + (5.59 - 3.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.83 - 2.36i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.401 + 1.5i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.19 + 5.19i)T - 53iT^{2} \)
59 \( 1 + (-1.56 + 0.901i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.6 + 7.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.33 + 1.96i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (10.1 - 10.1i)T - 73iT^{2} \)
79 \( 1 + (1.90 + 1.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.03 + 11.3i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.66T + 89T^{2} \)
97 \( 1 + (6.83 - 1.83i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.53322881165907007254024429802, −10.02318336044175037413083998187, −8.839567349531996546134480008016, −8.291191690533042840488286435997, −6.81742811777539554947099526481, −6.19930625641711831057795828605, −4.75026134602595625309753340228, −2.98648793544247946742369481533, −1.76483100445822218489445842634, 0, 2.80732246123771024690489243082, 4.25653121231433242872836760186, 5.76144197468747589146244677652, 6.42305718103961523420196390325, 7.04960419125320685872384802795, 8.696627690035223189385988551832, 9.523123169144815391493651059395, 10.17107827165779412002122047963, 10.74253914323615581440952599160

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