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Properties

Label 2-360-9.4-c1-0-6
Degree $2$
Conductor $360$
Sign $0.933 - 0.358i$
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  + (1.62 − 0.606i)3-s + (−0.5 + 0.866i)5-s + (2.62 + 4.54i)7-s + (2.26 − 1.96i)9-s + (−1.33 − 2.31i)11-s + (−1.90 + 3.30i)13-s + (−0.285 + 1.70i)15-s + 3.52·17-s − 4.67·19-s + (7.00 + 5.77i)21-s + (2.47 − 4.29i)23-s + (−0.499 − 0.866i)25-s + (2.48 − 4.56i)27-s + (0.928 + 1.60i)29-s + (4.33 − 7.51i)31-s + ⋯
L(s)  = 1  + (0.936 − 0.350i)3-s + (−0.223 + 0.387i)5-s + (0.991 + 1.71i)7-s + (0.754 − 0.655i)9-s + (−0.402 − 0.697i)11-s + (−0.529 + 0.916i)13-s + (−0.0738 + 0.441i)15-s + 0.855·17-s − 1.07·19-s + (1.52 + 1.26i)21-s + (0.516 − 0.895i)23-s + (−0.0999 − 0.173i)25-s + (0.477 − 0.878i)27-s + (0.172 + 0.298i)29-s + (0.778 − 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.933 - 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82945 + 0.338929i\)
\(L(\frac12)\) \(\approx\) \(1.82945 + 0.338929i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 0.606i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-2.62 - 4.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.90 - 3.30i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + (-2.47 + 4.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.928 - 1.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.33 + 7.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 + (-1.83 + 3.18i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.76 + 3.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.63 + 8.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.85T + 53T^{2} \)
59 \( 1 + (2.10 - 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.98 - 6.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.429 + 0.744i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 - 6.28T + 73T^{2} \)
79 \( 1 + (2.81 + 4.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.94 - 3.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
97 \( 1 + (-1.91 - 3.32i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.75245175904221536404784856388, −10.60332149199487401420978828927, −9.370515286003464207888909102874, −8.537291283443477094774994930384, −8.065197574087808681063322032758, −6.82864904086196552248286628299, −5.70118648257996675860530200226, −4.43042167919994180223881376668, −2.87188052615094909976938975548, −2.03622774309509672050065634636, 1.44552161298896703227627031468, 3.20806757218459966568254690268, 4.41094697396850515100216317704, 5.00812274795593065185813647777, 7.05861184820806172923690156181, 7.81702681780078742305705611775, 8.299591938021576109344783424651, 9.749820903908420698910347024531, 10.32906063489754579637168900142, 11.11357463299961939158155160950

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