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Properties

Label 2-360-40.29-c1-0-17
Degree $2$
Conductor $360$
Sign $-0.632 + 0.774i$
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  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−1.41 + 1.73i)5-s − 2.44i·7-s + 2.82·8-s + (3.12 + 0.507i)10-s − 3.46i·11-s + (−2.99 + 1.73i)14-s + (−2.00 − 3.46i)16-s − 4.89i·17-s − 3.46i·19-s + (−1.58 − 4.18i)20-s + (−4.24 + 2.44i)22-s − 2.44i·23-s + (−0.999 − 4.89i)25-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.632 + 0.774i)5-s − 0.925i·7-s + 0.999·8-s + (0.987 + 0.160i)10-s − 1.04i·11-s + (−0.801 + 0.462i)14-s + (−0.500 − 0.866i)16-s − 1.18i·17-s − 0.794i·19-s + (−0.354 − 0.935i)20-s + (−0.904 + 0.522i)22-s − 0.510i·23-s + (−0.199 − 0.979i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301946 - 0.636348i\)
\(L(\frac12)\) \(\approx\) \(0.301946 - 0.636348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 \)
5 \( 1 + (1.41 - 1.73i)T \)
good7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 2.44iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 - 7.34iT - 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−11.05067090628752658587391102611, −10.46211859376596822964885300349, −9.415201003408064518846219127462, −8.357016370286650825145375712662, −7.48534017911455403144071472684, −6.64641319604065193482331579144, −4.77462818805727155274658996389, −3.64399532427296982381799044059, −2.73956007242118671525097618182, −0.59181828826030600195408820638, 1.71115904636216459061935148406, 4.00230471430741844087964481399, 5.09215900525180567985896061918, 5.97418809060218922823645102518, 7.20312082287549614842798250597, 8.140052446631286621262658391081, 8.782643104203805176978297195768, 9.676407889636914983407309321067, 10.61608955293877426336405952147, 11.97159458937767749741061402003

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