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Properties

Label 2-360-360.173-c1-0-54
Degree $2$
Conductor $360$
Sign $0.280 + 0.959i$
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  + (0.105 + 1.41i)2-s + (0.853 − 1.50i)3-s + (−1.97 + 0.297i)4-s + (−2.16 + 0.550i)5-s + (2.21 + 1.04i)6-s + (−0.556 − 2.07i)7-s + (−0.628 − 2.75i)8-s + (−1.54 − 2.57i)9-s + (−1.00 − 2.99i)10-s + (2.27 − 3.94i)11-s + (−1.24 + 3.23i)12-s + (−5.01 − 1.34i)13-s + (2.87 − 1.00i)14-s + (−1.02 + 3.73i)15-s + (3.82 − 1.17i)16-s + (1.47 + 1.47i)17-s + ⋯
L(s)  = 1  + (0.0746 + 0.997i)2-s + (0.493 − 0.870i)3-s + (−0.988 + 0.148i)4-s + (−0.969 + 0.246i)5-s + (0.904 + 0.426i)6-s + (−0.210 − 0.785i)7-s + (−0.222 − 0.974i)8-s + (−0.513 − 0.857i)9-s + (−0.317 − 0.948i)10-s + (0.685 − 1.18i)11-s + (−0.358 + 0.933i)12-s + (−1.39 − 0.372i)13-s + (0.767 − 0.268i)14-s + (−0.263 + 0.964i)15-s + (0.955 − 0.294i)16-s + (0.357 + 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697453 - 0.522803i\)
\(L(\frac12)\) \(\approx\) \(0.697453 - 0.522803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.105 - 1.41i)T \)
3 \( 1 + (-0.853 + 1.50i)T \)
5 \( 1 + (2.16 - 0.550i)T \)
good7 \( 1 + (0.556 + 2.07i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.27 + 3.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.01 + 1.34i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.47 - 1.47i)T + 17iT^{2} \)
19 \( 1 + 3.52T + 19T^{2} \)
23 \( 1 + (-5.35 - 1.43i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.03 - 0.596i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.03 + 1.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.73 + 7.73i)T + 37iT^{2} \)
41 \( 1 + (0.729 - 0.420i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.174 + 0.651i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.73 - 0.733i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.72 - 2.72i)T + 53iT^{2} \)
59 \( 1 + (-8.67 + 5.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.42 - 4.28i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.58 - 13.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.60iT - 71T^{2} \)
73 \( 1 + (3.05 + 3.05i)T + 73iT^{2} \)
79 \( 1 + (-2.76 - 1.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.40 - 1.44i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 0.811T + 89T^{2} \)
97 \( 1 + (4.42 + 16.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.42959927372223127578375500866, −10.20384226211610170726060450176, −8.909440106913474756562032371619, −8.271870920176502151698616897436, −7.22633168106536678555526904697, −6.93853577441831189823326017087, −5.63372113649977132601933416635, −4.09041935284017829056170176823, −3.19570594442510810834319862001, −0.55330349889434504654911984800, 2.24845697778078111209027485278, 3.39224401603896822658710023039, 4.56660486005591213618467572416, 5.04565281330745226081271720276, 7.07079638900463796604553395428, 8.341476047792480487484421832004, 9.102464892716887126340845726402, 9.759509456143141372975654200749, 10.66209720274002255289725524990, 11.85551951234868435571977008484

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