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Properties

Label 2-360-15.8-c1-0-5
Degree $2$
Conductor $360$
Sign $-0.920 + 0.391i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.920 + 0.391i$
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: \((2,\ 360,\ (\ :1/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0601552 - 0.295433i\)
\(L(\frac12)\) \(\approx\) \(0.0601552 - 0.295433i\)
\(L(\frac{3}{2})\) 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

−11.23230896268049738518850690651, −9.941634226106017187892066929279, −9.139799403536941391739292000451, −8.349694305188907525065796428577, −7.38011629283075409797108977475, −5.90930865142238644106957914924, −5.36486095394586501043088152620, −3.72005732737048326520348541771, −2.70535345296119346195760227438, −0.19192887249395921764545045393, 2.42304487944256589698408059732, 3.88619116439759465582619811756, 4.55095635919019304107887978581, 6.59358665638736290789644875054, 6.97062901112379871399801037949, 7.77760627927444652439939341996, 9.403381347867565213111832383922, 9.978764618418999912830717234990, 10.82516942753701372802113552082, 11.87576114195678646289669046870

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