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Properties

Label 2-360-40.29-c1-0-15
Degree $2$
Conductor $360$
Sign $0.884 - 0.466i$
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  + (1.34 + 0.450i)2-s + (1.59 + 1.20i)4-s + (0.254 − 2.22i)5-s + 2.64i·7-s + (1.59 + 2.33i)8-s + (1.34 − 2.86i)10-s + 1.51i·11-s + 3.87·13-s + (−1.18 + 3.54i)14-s + (1.08 + 3.84i)16-s − 3.31i·17-s − 7.08i·19-s + (3.08 − 3.23i)20-s + (−0.681 + 2.02i)22-s + 4.82i·23-s + ⋯
L(s)  = 1  + (0.947 + 0.318i)2-s + (0.797 + 0.603i)4-s + (0.113 − 0.993i)5-s + 0.998i·7-s + (0.563 + 0.825i)8-s + (0.423 − 0.905i)10-s + 0.456i·11-s + 1.07·13-s + (−0.317 + 0.946i)14-s + (0.271 + 0.962i)16-s − 0.803i·17-s − 1.62i·19-s + (0.690 − 0.723i)20-s + (−0.145 + 0.432i)22-s + 1.00i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.884 - 0.466i$
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: \((2,\ 360,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34236 + 0.579599i\)
\(L(\frac12)\) \(\approx\) \(2.34236 + 0.579599i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.450i)T \)
3 \( 1 \)
5 \( 1 + (-0.254 + 2.22i)T \)
good7 \( 1 - 2.64iT - 7T^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 - 3.87T + 13T^{2} \)
17 \( 1 + 3.31iT - 17T^{2} \)
19 \( 1 + 7.08iT - 19T^{2} \)
23 \( 1 - 4.82iT - 23T^{2} \)
29 \( 1 + 2.18iT - 29T^{2} \)
31 \( 1 + 7.36T + 31T^{2} \)
37 \( 1 + 7.87T + 37T^{2} \)
41 \( 1 + 8.72T + 41T^{2} \)
43 \( 1 - 1.01T + 43T^{2} \)
47 \( 1 - 7.08iT - 47T^{2} \)
53 \( 1 + 4.50T + 53T^{2} \)
59 \( 1 + 6.79iT - 59T^{2} \)
61 \( 1 + 3.60iT - 61T^{2} \)
67 \( 1 + 1.01T + 67T^{2} \)
71 \( 1 - 6.72T + 71T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−11.75943404451251817889144275284, −10.99803774029757816113269671476, −9.368776003603063732987365085141, −8.755068719934589210053380101282, −7.64827077759182711522512887231, −6.50337637694266730441953605665, −5.41877506110159868588410946213, −4.82974114920931660372560045327, −3.43502355762640642153229140711, −1.96040682753129733989406636289, 1.70717648325546091317542950325, 3.43813584515417679861540403641, 3.92126312093008078809157971821, 5.59010442374956402349068876320, 6.43139867470728115367100569362, 7.25499631489612752313443464816, 8.452585551978429721306699237481, 10.18924517908408120807203854532, 10.52769495949567345566664475919, 11.25325662640779636555625584702

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