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Properties

Label 2-180-180.119-c1-0-24
Degree $2$
Conductor $180$
Sign $0.447 + 0.894i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (0.0700 + 1.41i)2-s + (−0.424 − 1.67i)3-s + (−1.99 + 0.197i)4-s + (−1.98 − 1.03i)5-s + (2.34 − 0.717i)6-s + (1.60 − 2.78i)7-s + (−0.419 − 2.79i)8-s + (−2.63 + 1.42i)9-s + (1.32 − 2.87i)10-s + (1.56 − 2.70i)11-s + (1.17 + 3.25i)12-s + (1.59 − 0.923i)13-s + (4.04 + 2.07i)14-s + (−0.892 + 3.76i)15-s + (3.92 − 0.787i)16-s − 5.85·17-s + ⋯
L(s)  = 1  + (0.0495 + 0.998i)2-s + (−0.245 − 0.969i)3-s + (−0.995 + 0.0989i)4-s + (−0.886 − 0.462i)5-s + (0.956 − 0.292i)6-s + (0.608 − 1.05i)7-s + (−0.148 − 0.988i)8-s + (−0.879 + 0.475i)9-s + (0.417 − 0.908i)10-s + (0.471 − 0.816i)11-s + (0.340 + 0.940i)12-s + (0.443 − 0.256i)13-s + (1.08 + 0.555i)14-s + (−0.230 + 0.973i)15-s + (0.980 − 0.196i)16-s − 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666558 - 0.412055i\)
\(L(\frac12)\) \(\approx\) \(0.666558 - 0.412055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0700 - 1.41i)T \)
3 \( 1 + (0.424 + 1.67i)T \)
5 \( 1 + (1.98 + 1.03i)T \)
good7 \( 1 + (-1.60 + 2.78i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.56 + 2.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.59 + 0.923i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.85T + 17T^{2} \)
19 \( 1 + 2.24iT - 19T^{2} \)
23 \( 1 + (3.24 - 1.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.90 - 3.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.03 + 0.599i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.66iT - 37T^{2} \)
41 \( 1 + (0.208 - 0.120i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.66 + 4.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.96 + 2.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + (3.47 + 6.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.66 + 2.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.31 + 7.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + (-11.6 - 6.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.07 + 0.623i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (5.75 + 3.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.73034910723226611660469426873, −11.59933695674534325182755707249, −10.75287016057723419638023759522, −8.824759699213837409007235214754, −8.222088572310083687836341756600, −7.27426903916726574183900025763, −6.43980199877218504317717967190, −5.01877796484159971978364932103, −3.84751364488421543991339493811, −0.77060375917625361019707359518, 2.49217310912488119194129785936, 4.03078450262012867413484842058, 4.74464298678333959715878605742, 6.26771312213612504869844066811, 8.246803469342799533420591561260, 8.951177791155430411444372757827, 10.07994742681521218206099432681, 11.00372157742497386461378312274, 11.79357322859145547151223033324, 12.22083628363765737609431486852

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