Properties

Label 2-360-45.4-c1-0-4
Degree $2$
Conductor $360$
Sign $-0.00456 - 0.999i$
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  + (0.284 + 1.70i)3-s + (1.07 − 1.96i)5-s + (−3.55 + 2.05i)7-s + (−2.83 + 0.973i)9-s + (3.04 + 5.27i)11-s + (4.45 + 2.56i)13-s + (3.65 + 1.27i)15-s + 2.73i·17-s − 1.80·19-s + (−4.51 − 5.48i)21-s + (−0.582 − 0.336i)23-s + (−2.69 − 4.20i)25-s + (−2.47 − 4.57i)27-s + (1.75 + 3.03i)29-s + (3.30 − 5.71i)31-s + ⋯
L(s)  = 1  + (0.164 + 0.986i)3-s + (0.479 − 0.877i)5-s + (−1.34 + 0.775i)7-s + (−0.945 + 0.324i)9-s + (0.918 + 1.59i)11-s + (1.23 + 0.712i)13-s + (0.944 + 0.328i)15-s + 0.663i·17-s − 0.414·19-s + (−0.985 − 1.19i)21-s + (−0.121 − 0.0701i)23-s + (−0.539 − 0.841i)25-s + (−0.475 − 0.879i)27-s + (0.325 + 0.563i)29-s + (0.592 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00456 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00456 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.00456 - 0.999i$
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: \((2,\ 360,\ (\ :1/2),\ -0.00456 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924361 + 0.928586i\)
\(L(\frac12)\) \(\approx\) \(0.924361 + 0.928586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.284 - 1.70i)T \)
5 \( 1 + (-1.07 + 1.96i)T \)
good7 \( 1 + (3.55 - 2.05i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.04 - 5.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.45 - 2.56i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.73iT - 17T^{2} \)
19 \( 1 + 1.80T + 19T^{2} \)
23 \( 1 + (0.582 + 0.336i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.30 + 5.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.44iT - 37T^{2} \)
41 \( 1 + (2.08 - 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.01 + 2.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.38 + 0.798i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.02iT - 53T^{2} \)
59 \( 1 + (2.65 - 4.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.82 + 5.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 6.50iT - 73T^{2} \)
79 \( 1 + (-1.49 - 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.71 + 2.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.19T + 89T^{2} \)
97 \( 1 + (-3.33 + 1.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.79753383218153756004882595585, −10.48860054463226720230273775571, −9.510214204304667578746376810321, −9.257301758624367988318951699391, −8.373262803619258941353568293807, −6.56251501647529674433286199609, −5.88715374148153192043152461384, −4.56602870763984879735267632306, −3.72817355666531284636102902345, −2.07420166562942089741520987741, 0.930028073742326162486359929528, 2.97659208663223197951642762524, 3.54975871605751760932763896461, 6.05132975226259176641182357407, 6.27183143899212100028519272151, 7.19332592184841394344826272233, 8.391033321943252404031990665024, 9.289723604844849537911829035048, 10.45979455450419269810632093988, 11.13081347366305977189991900194

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