Properties

Label 2-360-45.4-c1-0-9
Degree $2$
Conductor $360$
Sign $0.595 - 0.803i$
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  + (1.19 + 1.25i)3-s + (1.61 + 1.54i)5-s + (1.98 − 1.14i)7-s + (−0.139 + 2.99i)9-s + (0.864 + 1.49i)11-s + (−5.20 − 3.00i)13-s + (−0.0131 + 3.87i)15-s − 4.65i·17-s − 0.888·19-s + (3.80 + 1.11i)21-s + (5.17 + 2.98i)23-s + (0.198 + 4.99i)25-s + (−3.92 + 3.40i)27-s + (−2.34 − 4.06i)29-s + (−2.98 + 5.16i)31-s + ⋯
L(s)  = 1  + (0.690 + 0.723i)3-s + (0.721 + 0.692i)5-s + (0.748 − 0.432i)7-s + (−0.0464 + 0.998i)9-s + (0.260 + 0.451i)11-s + (−1.44 − 0.832i)13-s + (−0.00339 + 0.999i)15-s − 1.12i·17-s − 0.203·19-s + (0.829 + 0.243i)21-s + (1.07 + 0.623i)23-s + (0.0397 + 0.999i)25-s + (−0.754 + 0.656i)27-s + (−0.435 − 0.754i)29-s + (−0.535 + 0.927i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.595 - 0.803i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69094 + 0.851810i\)
\(L(\frac12)\) \(\approx\) \(1.69094 + 0.851810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.19 - 1.25i)T \)
5 \( 1 + (-1.61 - 1.54i)T \)
good7 \( 1 + (-1.98 + 1.14i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.864 - 1.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.20 + 3.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.65iT - 17T^{2} \)
19 \( 1 + 0.888T + 19T^{2} \)
23 \( 1 + (-5.17 - 2.98i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.34 + 4.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.98 - 5.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.68iT - 37T^{2} \)
41 \( 1 + (3.15 - 5.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.46 + 0.843i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.8 + 6.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.66iT - 53T^{2} \)
59 \( 1 + (-1.62 + 2.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.08 + 10.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.8 + 6.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.25T + 71T^{2} \)
73 \( 1 + 1.69iT - 73T^{2} \)
79 \( 1 + (1.99 + 3.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.63 - 2.09i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.60T + 89T^{2} \)
97 \( 1 + (3.46 - 1.99i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.24777645075448872053230118306, −10.55119776911636840641962393762, −9.717843300348272550087496638936, −9.098808726936837880406404071352, −7.63211315060332806612843134001, −7.18789614723197440499180176702, −5.43445062869368891308554723361, −4.68039464667593558278895011166, −3.20931430859298008336777719463, −2.14307726972830063280150179154, 1.51823887383352704876690106938, 2.53766501624749086388585316925, 4.30390741186269254943109269858, 5.47537792734625192335655773566, 6.57009861329077319924206029255, 7.62785309159604948641966521174, 8.733373690866038966955920710467, 9.067716640854842391470408460811, 10.24034623748599074378249844763, 11.55373849575069259023812821753

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