Properties

Label 2-360-15.8-c1-0-1
Degree $2$
Conductor $360$
Sign $-0.0618 - 0.998i$
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  + (1.58 + 1.58i)5-s + (−3.23 + 3.23i)7-s + 4.57i·11-s + (−4.23 − 4.23i)13-s + (1.74 + 1.74i)17-s + 2.47i·19-s + (2.82 − 2.82i)23-s + 5.00i·25-s + 5.99·29-s + 1.52·31-s − 10.2·35-s + (2.23 − 2.23i)37-s + 7.07i·41-s + (2.47 + 2.47i)43-s + (−1.74 − 1.74i)47-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)5-s + (−1.22 + 1.22i)7-s + 1.37i·11-s + (−1.17 − 1.17i)13-s + (0.423 + 0.423i)17-s + 0.567i·19-s + (0.589 − 0.589i)23-s + 1.00i·25-s + 1.11·29-s + 0.274·31-s − 1.72·35-s + (0.367 − 0.367i)37-s + 1.10i·41-s + (0.376 + 0.376i)43-s + (−0.254 − 0.254i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0618 - 0.998i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.0618 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796680 + 0.847591i\)
\(L(\frac12)\) \(\approx\) \(0.796680 + 0.847591i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.58 - 1.58i)T \)
good7 \( 1 + (3.23 - 3.23i)T - 7iT^{2} \)
11 \( 1 - 4.57iT - 11T^{2} \)
13 \( 1 + (4.23 + 4.23i)T + 13iT^{2} \)
17 \( 1 + (-1.74 - 1.74i)T + 17iT^{2} \)
19 \( 1 - 2.47iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 5.99T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 + (-2.23 + 2.23i)T - 37iT^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (-2.47 - 2.47i)T + 43iT^{2} \)
47 \( 1 + (1.74 + 1.74i)T + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 1.08T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + (1.52 - 1.52i)T - 67iT^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + (0.527 + 0.527i)T + 73iT^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 + (-1.08 + 1.08i)T - 83iT^{2} \)
89 \( 1 - 0.746T + 89T^{2} \)
97 \( 1 + (-1 + i)T - 97iT^{2} \)
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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

−11.93586441804593990388983897204, −10.33350984726477638000187493985, −9.975452548995130587311736026490, −9.196790076038281305302103272540, −7.84649566468525447246973160436, −6.76493149300548628641919907680, −5.97170564025349804095343286354, −4.96266267944122275292917979590, −3.10086776286565307561107331628, −2.33379078133200493294832038044, 0.793328701528245584417477695339, 2.81504521716507621262939527475, 4.12758134964691048001470966984, 5.30405227627593869601132994577, 6.48926497799519694047581823726, 7.20433124287065987305798816113, 8.601085493089828240476944051781, 9.521895035556398647843320202295, 10.04246955225837172549462345993, 11.15993933843407661520071790005

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