Properties

Label 2-360-9.4-c1-0-7
Degree $2$
Conductor $360$
Sign $0.986 + 0.164i$
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  + (1.72 − 0.158i)3-s + (0.5 − 0.866i)5-s + (0.724 + 1.25i)7-s + (2.94 − 0.548i)9-s + (0.724 − 1.57i)15-s − 2·17-s + 2.89·19-s + (1.44 + 2.04i)21-s + (1.27 − 2.20i)23-s + (−0.499 − 0.866i)25-s + (4.99 − 1.41i)27-s + (−3.94 − 6.84i)29-s + (−5.44 + 9.43i)31-s + 1.44·35-s − 6·37-s + ⋯
L(s)  = 1  + (0.995 − 0.0917i)3-s + (0.223 − 0.387i)5-s + (0.273 + 0.474i)7-s + (0.983 − 0.182i)9-s + (0.187 − 0.406i)15-s − 0.485·17-s + 0.665·19-s + (0.316 + 0.447i)21-s + (0.265 − 0.460i)23-s + (−0.0999 − 0.173i)25-s + (0.962 − 0.272i)27-s + (−0.733 − 1.27i)29-s + (−0.978 + 1.69i)31-s + 0.245·35-s − 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.986 + 0.164i$
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: \((2,\ 360,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94747 - 0.161330i\)
\(L(\frac12)\) \(\approx\) \(1.94747 - 0.161330i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.158i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-0.724 - 1.25i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 2.89T + 19T^{2} \)
23 \( 1 + (-1.27 + 2.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.94 + 6.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.44 - 9.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-0.0505 + 0.0874i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.89 - 6.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.27 - 3.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.62 - 9.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.79T + 71T^{2} \)
73 \( 1 + 5.79T + 73T^{2} \)
79 \( 1 + (1.44 + 2.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.275 - 0.476i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.48030508330562746998505522395, −10.33025784353295685360432515962, −9.330842616536533941992812676537, −8.737107781005639933990326748194, −7.81992017963868845293862917471, −6.80474761404584793533608593350, −5.46436336555606318863860802678, −4.31804690430714939877816087568, −2.98189148802267898869008997397, −1.69963129427529424521091621478, 1.80144596751132884212957624817, 3.18251269739172063177625062524, 4.21677105280676442361984230932, 5.56119023600477559060575936366, 7.07107515254452554170514653658, 7.58033622860044370313005924892, 8.803629329622636128334258934669, 9.511493626184509553261518477049, 10.49969951836145950144410732875, 11.26024076594844072621895541245

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