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Properties

Label 2-219-219.137-c0-0-2
Degree $2$
Conductor $219$
Sign $0.373 + 0.927i$
Analytic cond. $0.109295$
Root an. cond. $0.330598$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s − i·3-s + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)6-s + i·8-s − 9-s − 0.999·10-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + 0.999·22-s + (−0.866 − 0.5i)23-s + 24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s − i·3-s + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)6-s + i·8-s − 9-s − 0.999·10-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + 0.999·22-s + (−0.866 − 0.5i)23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 219 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(219\)    =    \(3 \cdot 73\)
Sign: $0.373 + 0.927i$
Analytic conductor: \(0.109295\)
Root analytic conductor: \(0.330598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{219} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 219,\ (\ :0),\ 0.373 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9120084735\)
\(L(\frac12)\) \(\approx\) \(0.9120084735\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
73 \( 1 + T \)
good2 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−12.39538625562174324670897388737, −11.70427742637163161584900095702, −11.10431622177311561220989381147, −9.115061166956220937802536284317, −8.346867242852283231094065261921, −7.29335706090320299408775243351, −6.12431870987516470758889059761, −4.62468068122705460477081580654, −3.72625818105839647164751979392, −2.04450464346036654471394195150, 3.58492594127133134992539910688, 3.93142438706968218785157178014, 5.43035803916290434209300285170, 6.21560617127001548551881573055, 7.60409008150833345560627439468, 8.784299478824379044515155798543, 9.925086327081380720047234201942, 10.83392034233258331996764182900, 11.70579172944038568760184127245, 12.80747924072578978692322372089

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