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Properties

Label 2-360-40.29-c1-0-23
Degree $2$
Conductor $360$
Sign $0.395 + 0.918i$
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  + (1.11 − 0.866i)2-s + (0.500 − 1.93i)4-s + 2.23·5-s + (−1.11 − 2.59i)8-s + (2.50 − 1.93i)10-s + (−3.5 − 1.93i)16-s + 6.92i·17-s − 7.74i·19-s + (1.11 − 4.33i)20-s + 3.46i·23-s + 5.00·25-s − 8·31-s + (−5.59 + 0.866i)32-s + (5.99 + 7.74i)34-s + (−6.70 − 8.66i)38-s + ⋯
L(s)  = 1  + (0.790 − 0.612i)2-s + (0.250 − 0.968i)4-s + 0.999·5-s + (−0.395 − 0.918i)8-s + (0.790 − 0.612i)10-s + (−0.875 − 0.484i)16-s + 1.68i·17-s − 1.77i·19-s + (0.250 − 0.968i)20-s + 0.722i·23-s + 1.00·25-s − 1.43·31-s + (−0.988 + 0.153i)32-s + (1.02 + 1.32i)34-s + (−1.08 − 1.40i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.395 + 0.918i$
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: \((2,\ 360,\ (\ :1/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91827 - 1.26285i\)
\(L(\frac12)\) \(\approx\) \(1.91827 - 1.26285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.866i)T \)
3 \( 1 \)
5 \( 1 - 2.23T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6.92iT - 17T^{2} \)
19 \( 1 + 7.74iT - 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 + 4.47T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 15.4iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 17.8T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.13499021743114807865870147567, −10.61267511970027455883625139022, −9.584708606574085658752453678346, −8.843517546451954994235362663959, −7.17167154835871913373277622081, −6.14079877240387661532996606886, −5.36982727164323878418020054903, −4.19008235000124739392123937359, −2.82379646482041172845711512995, −1.58583667890859363804656841915, 2.19089763468231543384921930973, 3.52764951586358401703680591064, 4.94119399733646795821694323111, 5.71532054223491841891359507798, 6.67632237648738808642492125597, 7.62034062224646169049964955014, 8.758879710799035024927040166656, 9.694454674644094737359607981509, 10.76111173788021998835451595633, 11.88954589695871451672948545080

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