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Properties

Label 2-360-8.5-c1-0-19
Degree $2$
Conductor $360$
Sign $-0.707 + 0.707i$
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 − i)2-s − 2i·4-s − i·5-s − 4·7-s + (−2 − 2i)8-s + (−1 − i)10-s − 2i·11-s − 6i·13-s + (−4 + 4i)14-s − 4·16-s + 6·17-s + 4i·19-s − 2·20-s + (−2 − 2i)22-s + 8·23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s − 0.447i·5-s − 1.51·7-s + (−0.707 − 0.707i)8-s + (−0.316 − 0.316i)10-s − 0.603i·11-s − 1.66i·13-s + (−1.06 + 1.06i)14-s − 16-s + 1.45·17-s + 0.917i·19-s − 0.447·20-s + (−0.426 − 0.426i)22-s + 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586276 - 1.41539i\)
\(L(\frac12)\) \(\approx\) \(0.586276 - 1.41539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 - 2iT - 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−11.08167584125155561638765995531, −10.21898822244366076901623566281, −9.609472923568315897006149497409, −8.492225997074432379526711562821, −7.09247381291939063699403692344, −5.82301876870837186163342380281, −5.32179651590513290569994241952, −3.53986017831255650127338570475, −3.06198098949176090838714246157, −0.865342802778594093176812575350, 2.70689072658095983430423910971, 3.71954981721229800996482353866, 4.90918623907008113969976221393, 6.26455237169498505723711878876, 6.80932736559196510284343795770, 7.63696345863375048975036301236, 9.176200039481152561401293759703, 9.632688460141725728316925891017, 11.07695845840639969512774657759, 12.03044852355822776971956849523

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