Properties

Label 2-360-45.34-c1-0-0
Degree $2$
Conductor $360$
Sign $-0.966 + 0.257i$
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 + (−2.18 − 0.487i)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 + (−0.416 − 3.85i)15-s + 4.59i·17-s − 6.38·19-s + (1.51 − 7.65i)21-s + (5.34 − 3.08i)23-s + (4.52 + 2.12i)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 + (−0.975 − 0.218i)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 + (−0.107 − 0.994i)15-s + 1.11i·17-s − 1.46·19-s + (0.331 − 1.67i)21-s + (1.11 − 0.642i)23-s + (0.904 + 0.425i)25-s + (−0.832 − 0.554i)27-s + (0.362 − 0.628i)29-s + (0.194 + 0.337i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.966 + 0.257i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.966 + 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0201960 - 0.154013i\)
\(L(\frac12)\) \(\approx\) \(0.0201960 - 0.154013i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.487i)T \)
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

−11.91877673569822506627561481238, −10.71903676130505875261197417119, −10.22985714147554130872780298283, −9.167185187965778943658890526647, −8.432230149195811655722633764256, −7.21285551529371279223585510828, −6.29056436774148250290684753598, −4.59334622570862838962974627733, −3.97137357170745355061546769006, −2.90842271387462293136179937133, 0.094912842253312430140220370829, 2.58333910628983544338462956238, 3.31782494199050439554113822212, 5.05552225919538483041338811291, 6.48584670557253149998958640896, 7.03743283709576863223372490092, 8.032265539902579066719133674341, 9.035502578374753998914486167998, 9.822388453561050355246542609623, 11.19799653939155989368621113644

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