Properties

Label 2-360-45.4-c1-0-6
Degree $2$
Conductor $360$
Sign $0.176 - 0.984i$
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  + (−0.557 + 1.63i)3-s + (0.668 + 2.13i)5-s + (3.90 − 2.25i)7-s + (−2.37 − 1.82i)9-s + (0.631 + 1.09i)11-s + (4.31 + 2.48i)13-s + (−3.87 − 0.0927i)15-s + 4.59i·17-s − 6.38·19-s + (1.51 + 7.65i)21-s + (−5.34 − 3.08i)23-s + (−4.10 + 2.85i)25-s + (4.32 − 2.88i)27-s + (1.95 + 3.38i)29-s + (1.08 − 1.88i)31-s + ⋯
L(s)  = 1  + (−0.321 + 0.946i)3-s + (0.299 + 0.954i)5-s + (1.47 − 0.851i)7-s + (−0.792 − 0.609i)9-s + (0.190 + 0.329i)11-s + (1.19 + 0.690i)13-s + (−0.999 − 0.0239i)15-s + 1.11i·17-s − 1.46·19-s + (0.331 + 1.67i)21-s + (−1.11 − 0.642i)23-s + (−0.821 + 0.570i)25-s + (0.832 − 0.554i)27-s + (0.362 + 0.628i)29-s + (0.194 − 0.337i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.176 - 0.984i$
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: \((2,\ 360,\ (\ :1/2),\ 0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09101 + 0.913142i\)
\(L(\frac12)\) \(\approx\) \(1.09101 + 0.913142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.557 - 1.63i)T \)
5 \( 1 + (-0.668 - 2.13i)T \)
good7 \( 1 + (-3.90 + 2.25i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.631 - 1.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.31 - 2.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.59iT - 17T^{2} \)
19 \( 1 + 6.38T + 19T^{2} \)
23 \( 1 + (5.34 + 3.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.95 - 3.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.08 + 1.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.516iT - 37T^{2} \)
41 \( 1 + (-1.11 + 1.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.09 + 2.94i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.95 + 3.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.92iT - 53T^{2} \)
59 \( 1 + (-3.91 + 6.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.62 + 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.63 - 1.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.06T + 71T^{2} \)
73 \( 1 + 6.90iT - 73T^{2} \)
79 \( 1 + (-4.09 - 7.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.71 - 3.29i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.969T + 89T^{2} \)
97 \( 1 + (-10.8 + 6.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.15689291770361347880801014426, −10.78166319646973480619937333595, −10.21367807243780463576460191310, −8.854115714353593364541804276393, −8.044733663169592601197059056521, −6.68560486897648094145498361158, −5.88543780871316713960389935683, −4.38447499526963255647251223762, −3.88252320753449986751986603101, −1.92161962755715645809657239273, 1.17432321633510179580476890087, 2.33899106459246002560299857120, 4.47052623483703485537024841694, 5.56267377198875240455879034985, 6.11741003148097754807101054015, 7.74199057098810017750636215706, 8.391627071945268799256561951282, 8.979492810238172962733971227165, 10.57342578905650624625909800179, 11.55766438961929679314226722112

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