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Properties

Label 2-2184-13.3-c1-0-23
Degree $2$
Conductor $2184$
Sign $-0.234 + 0.972i$
Analytic cond. $17.4393$
Root an. cond. $4.17604$
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  + (−0.5 − 0.866i)3-s − 2.89·5-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.0379 + 0.0658i)11-s + (3.60 − 0.0658i)13-s + (1.44 + 2.50i)15-s + (−0.0276 + 0.0479i)17-s + (0.0975 − 0.168i)19-s − 0.999·21-s + (1.97 + 3.41i)23-s + 3.37·25-s + 0.999·27-s + (0.716 + 1.24i)29-s + 5.66·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 1.29·5-s + (0.188 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.0114 + 0.0198i)11-s + (0.999 − 0.0182i)13-s + (0.373 + 0.646i)15-s + (−0.00671 + 0.0116i)17-s + (0.0223 − 0.0387i)19-s − 0.218·21-s + (0.411 + 0.713i)23-s + 0.674·25-s + 0.192·27-s + (0.133 + 0.230i)29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2184\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.234 + 0.972i$
Analytic conductor: \(17.4393\)
Root analytic conductor: \(4.17604\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2184} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2184,\ (\ :1/2),\ -0.234 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9689633493\)
\(L(\frac12)\) \(\approx\) \(0.9689633493\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-3.60 + 0.0658i)T \)
good5 \( 1 + 2.89T + 5T^{2} \)
11 \( 1 + (-0.0379 - 0.0658i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.0276 - 0.0479i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0975 + 0.168i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.97 - 3.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.716 - 1.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.66T + 31T^{2} \)
37 \( 1 + (4.34 + 7.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.213 - 0.370i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.05 + 1.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.156T + 47T^{2} \)
53 \( 1 + 8.24T + 53T^{2} \)
59 \( 1 + (-1.45 + 2.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.81 + 6.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.82 + 8.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.392 + 0.679i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 3.54T + 83T^{2} \)
89 \( 1 + (7.68 + 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.02 + 10.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.567099901034335152921288049486, −8.025322955284518979827076022423, −7.33592851147356724120084321990, −6.66330827265783517262597147595, −5.72752616933041029672937280408, −4.74794455956633998491087530551, −3.89125801165910276556448032995, −3.14118935825362141503072939406, −1.62951090670894543249557211606, −0.44124392934485279381974881349, 1.05479624550613179596502184666, 2.75759897162303588978431505461, 3.68312470112895631663153340424, 4.37846562659294095058110361851, 5.15573789977474396903984775057, 6.19689110708710623377317187150, 6.89041314051595389085677164484, 7.993522264475095079478239004532, 8.397261206907689136427023016983, 9.158036754360379932203546553490

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