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Properties

Label 2-360-40.27-c1-0-26
Degree $2$
Conductor $360$
Sign $-0.801 - 0.598i$
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  + (−0.788 − 1.17i)2-s + (−0.758 + 1.85i)4-s + (−0.386 − 2.20i)5-s + (−1.51 + 1.51i)7-s + (2.77 − 0.568i)8-s + (−2.28 + 2.18i)10-s − 3.92·11-s + (−3.56 − 3.56i)13-s + (2.97 + 0.585i)14-s + (−2.85 − 2.80i)16-s + (1.37 + 1.37i)17-s + 4i·19-s + (4.36 + 0.954i)20-s + (3.09 + 4.60i)22-s + (−5.17 − 5.17i)23-s + ⋯
L(s)  = 1  + (−0.557 − 0.830i)2-s + (−0.379 + 0.925i)4-s + (−0.172 − 0.984i)5-s + (−0.573 + 0.573i)7-s + (0.979 − 0.200i)8-s + (−0.721 + 0.692i)10-s − 1.18·11-s + (−0.988 − 0.988i)13-s + (0.795 + 0.156i)14-s + (−0.712 − 0.701i)16-s + (0.333 + 0.333i)17-s + 0.917i·19-s + (0.976 + 0.213i)20-s + (0.659 + 0.982i)22-s + (−1.07 − 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.801 - 0.598i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0592348 + 0.178359i\)
\(L(\frac12)\) \(\approx\) \(0.0592348 + 0.178359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.788 + 1.17i)T \)
3 \( 1 \)
5 \( 1 + (0.386 + 2.20i)T \)
good7 \( 1 + (1.51 - 1.51i)T - 7iT^{2} \)
11 \( 1 + 3.92T + 11T^{2} \)
13 \( 1 + (3.56 + 3.56i)T + 13iT^{2} \)
17 \( 1 + (-1.37 - 1.37i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (5.17 + 5.17i)T + 23iT^{2} \)
29 \( 1 + 5.95T + 29T^{2} \)
31 \( 1 - 7.12iT - 31T^{2} \)
37 \( 1 + (-3.56 + 3.56i)T - 37iT^{2} \)
41 \( 1 + 2.75T + 41T^{2} \)
43 \( 1 + (-5.40 + 5.40i)T - 43iT^{2} \)
47 \( 1 + (1.54 - 1.54i)T - 47iT^{2} \)
53 \( 1 + (1.81 + 1.81i)T + 53iT^{2} \)
59 \( 1 - 3.92iT - 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (5.40 + 5.40i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 + (-6.67 + 6.67i)T - 83iT^{2} \)
89 \( 1 + 18.4iT - 89T^{2} \)
97 \( 1 + (10.4 + 10.4i)T + 97iT^{2} \)
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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

−10.65843675240979509521498933554, −10.03351719704377756209954894312, −9.157356185197421425775295004018, −8.156423234217604524934145442561, −7.63691549908631972021343717819, −5.78549166246215239986602448687, −4.79657069709332408286360490503, −3.40765684882023671477181525948, −2.15031329907333678022553474693, −0.14312749591513282974425955231, 2.44087295412223992968608576010, 4.08990878349746906486700524469, 5.41363439555976680855780910771, 6.52977111561384750365147175707, 7.37753056819671570885301603044, 7.85506420335165765426082038300, 9.494743184294566569515703151201, 9.872383595581744919347391342142, 10.86937920835372767283976262019, 11.70016798252642551696196278223

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