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Properties

Label 2-360-24.11-c1-0-5
Degree $2$
Conductor $360$
Sign $-0.525 - 0.850i$
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  + (0.681 + 1.23i)2-s + (−1.07 + 1.68i)4-s + 5-s + 1.41i·7-s + (−2.82 − 0.175i)8-s + (0.681 + 1.23i)10-s + 6.37i·11-s − 3.54i·13-s + (−1.75 + 0.964i)14-s + (−1.70 − 3.61i)16-s + 3.92i·17-s + 1.27·19-s + (−1.07 + 1.68i)20-s + (−7.89 + 4.34i)22-s + 6.28·23-s + ⋯
L(s)  = 1  + (0.482 + 0.876i)2-s + (−0.535 + 0.844i)4-s + 0.447·5-s + 0.534i·7-s + (−0.998 − 0.0619i)8-s + (0.215 + 0.391i)10-s + 1.92i·11-s − 0.982i·13-s + (−0.468 + 0.257i)14-s + (−0.426 − 0.904i)16-s + 0.952i·17-s + 0.292·19-s + (−0.239 + 0.377i)20-s + (−1.68 + 0.925i)22-s + 1.31·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.525 - 0.850i$
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: \((2,\ 360,\ (\ :1/2),\ -0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.793827 + 1.42372i\)
\(L(\frac12)\) \(\approx\) \(0.793827 + 1.42372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.681 - 1.23i)T \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 - 6.37iT - 11T^{2} \)
13 \( 1 + 3.54iT - 13T^{2} \)
17 \( 1 - 3.92iT - 17T^{2} \)
19 \( 1 - 1.27T + 19T^{2} \)
23 \( 1 - 6.28T + 23T^{2} \)
29 \( 1 + 9.00T + 29T^{2} \)
31 \( 1 + 3.92iT - 31T^{2} \)
37 \( 1 + 2.51iT - 37T^{2} \)
41 \( 1 + 5.27iT - 41T^{2} \)
43 \( 1 - 1.55T + 43T^{2} \)
47 \( 1 - 9.73T + 47T^{2} \)
53 \( 1 - 5.55T + 53T^{2} \)
59 \( 1 - 0.313iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 + 7.00T + 67T^{2} \)
71 \( 1 + 0.990T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 8.18iT - 79T^{2} \)
83 \( 1 + 5.02iT - 83T^{2} \)
89 \( 1 - 0.386iT - 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−12.21073748633276887933479350986, −10.79352902996420200637114140331, −9.656311119849090769827092636990, −8.957164498341979396582168697380, −7.72605559165526206006486177472, −7.04666182253185737815682282431, −5.81391628355597829055086288547, −5.10978474655816132414590688472, −3.87592326277221873390286920757, −2.32439493219263104552664642614, 1.04825444583806461937419346741, 2.77504146160861910920654477091, 3.82187126917501070753920460209, 5.11713347030823826016667275002, 6.01772574654796768514878768774, 7.17785677418869663149500412538, 8.805668902747117254702106405792, 9.295342005896128010923598614755, 10.50319971195681397074374701354, 11.19032506031163010866493225843

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