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Properties

Label 2-180-180.67-c1-0-10
Degree $2$
Conductor $180$
Sign $0.992 - 0.121i$
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  + (−0.628 + 1.26i)2-s + (−1.72 + 0.119i)3-s + (−1.20 − 1.59i)4-s + (1.70 − 1.44i)5-s + (0.935 − 2.26i)6-s + (−1.23 − 0.331i)7-s + (2.77 − 0.529i)8-s + (2.97 − 0.411i)9-s + (0.753 + 3.07i)10-s + (4.09 − 2.36i)11-s + (2.27 + 2.60i)12-s + (0.448 + 1.67i)13-s + (1.19 − 1.35i)14-s + (−2.77 + 2.69i)15-s + (−1.07 + 3.85i)16-s + (−2.37 − 2.37i)17-s + ⋯
L(s)  = 1  + (−0.444 + 0.895i)2-s + (−0.997 + 0.0687i)3-s + (−0.604 − 0.796i)4-s + (0.763 − 0.645i)5-s + (0.382 − 0.924i)6-s + (−0.466 − 0.125i)7-s + (0.982 − 0.187i)8-s + (0.990 − 0.137i)9-s + (0.238 + 0.971i)10-s + (1.23 − 0.712i)11-s + (0.657 + 0.753i)12-s + (0.124 + 0.464i)13-s + (0.319 − 0.362i)14-s + (−0.717 + 0.696i)15-s + (−0.269 + 0.963i)16-s + (−0.575 − 0.575i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.992 - 0.121i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.992 - 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749292 + 0.0456883i\)
\(L(\frac12)\) \(\approx\) \(0.749292 + 0.0456883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.628 - 1.26i)T \)
3 \( 1 + (1.72 - 0.119i)T \)
5 \( 1 + (-1.70 + 1.44i)T \)
good7 \( 1 + (1.23 + 0.331i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.448 - 1.67i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.37 + 2.37i)T + 17iT^{2} \)
19 \( 1 - 2.26T + 19T^{2} \)
23 \( 1 + (-8.43 + 2.26i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.111 - 0.0641i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.75 + 3.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.433 + 0.433i)T + 37iT^{2} \)
41 \( 1 + (-3.06 + 5.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.09 - 4.10i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.63 - 1.24i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.98 - 3.98i)T - 53iT^{2} \)
59 \( 1 + (7.36 - 12.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.25 - 9.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0587 + 0.219i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.57iT - 71T^{2} \)
73 \( 1 + (4.90 - 4.90i)T - 73iT^{2} \)
79 \( 1 + (0.916 + 1.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.55 + 5.78i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (-0.803 + 3.00i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.87583173281623398503144862238, −11.56054305295178267640644245609, −10.57726135440086626119910887449, −9.317388258629229470499014472708, −9.018011403125999907108881672086, −7.15602068107163472297174631401, −6.35708977851489954251096472420, −5.47307273300422222435736068398, −4.32435404102053986564611500682, −1.06700095703977361590265897498, 1.61669764719405538524649994296, 3.43905095826251188888498242876, 5.00193502263115853255170237304, 6.42716076558079858488159329002, 7.29515651678947442173297269894, 9.136488016103892487191592468176, 9.783281303291772474844403450000, 10.79753750008392223019080466589, 11.41442382190254847281676884988, 12.59511508536213051704599278770

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