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Properties

Label 2-360-360.59-c1-0-34
Degree $2$
Conductor $360$
Sign $0.930 - 0.367i$
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  + (1.22 − 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 − 1.73i)4-s + (−1.67 + 1.48i)5-s + (2.12 + 1.22i)6-s + (1.22 + 2.12i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−1 + 3i)10-s + (4.5 − 2.59i)11-s + 3.46·12-s + (−1.22 + 2.12i)13-s + (3 + 1.73i)14-s + (−3.67 − 1.22i)15-s + (−2.00 − 3.46i)16-s + 5.19·17-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 + 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.748 + 0.663i)5-s + (0.866 + 0.499i)6-s + (0.462 + 0.801i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.316 + 0.948i)10-s + (1.35 − 0.783i)11-s + 0.999·12-s + (−0.339 + 0.588i)13-s + (0.801 + 0.462i)14-s + (−0.948 − 0.316i)15-s + (−0.500 − 0.866i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.930 - 0.367i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.930 - 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33950 + 0.445544i\)
\(L(\frac12)\) \(\approx\) \(2.33950 + 0.445544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (1.67 - 1.48i)T \)
good7 \( 1 + (-1.22 - 2.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 - 2.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-2.44 - 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.24 + 7.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.36 + 3.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.79 - 4.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.12 + 3.53i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.41iT - 53T^{2} \)
59 \( 1 + (1.5 + 0.866i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.36 + 3.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 3iT - 73T^{2} \)
79 \( 1 + (-6.36 + 3.67i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + (-2.59 + 1.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.52327650140586234543435856959, −10.88806816657939308978772023767, −9.765117375711090545878773111945, −8.929362662817609749017339201565, −7.79255956170418896054068557136, −6.44202779521117309826749938338, −5.40137025205246933603305790473, −4.12706058856715409499542821604, −3.50883867216640954807004189986, −2.21680169494069245525695379767, 1.52292669218635472072752445517, 3.42021677315747528070081684896, 4.26485628559014598223156763938, 5.46817253117397664984197094353, 6.94829247151600589916940270543, 7.34849349008979612180105719861, 8.290345685799207307849617978756, 9.119807496466927204893900121654, 10.79611395589134481917205033439, 11.86060527412480260159994931429

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