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Properties

Label 2-360-9.7-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: $\chi_{360} (241, \cdot )$
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

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

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