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Properties

Label 2-2163-2163.1511-c0-0-1
Degree $2$
Conductor $2163$
Sign $0.993 + 0.115i$
Analytic cond. $1.07947$
Root an. cond. $1.03897$
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  + (0.273 − 0.961i)3-s + (0.445 + 0.895i)4-s + (−0.932 + 0.361i)7-s + (−0.850 − 0.526i)9-s + (0.982 − 0.183i)12-s + (1.78 − 0.165i)13-s + (−0.602 + 0.798i)16-s + (0.694 − 0.197i)19-s + (0.0922 + 0.995i)21-s + (0.739 + 0.673i)25-s + (−0.739 + 0.673i)27-s + (−0.739 − 0.673i)28-s + (1.18 − 1.56i)31-s + (0.0922 − 0.995i)36-s + (0.365 + 1.95i)37-s + ⋯
L(s)  = 1  + (0.273 − 0.961i)3-s + (0.445 + 0.895i)4-s + (−0.932 + 0.361i)7-s + (−0.850 − 0.526i)9-s + (0.982 − 0.183i)12-s + (1.78 − 0.165i)13-s + (−0.602 + 0.798i)16-s + (0.694 − 0.197i)19-s + (0.0922 + 0.995i)21-s + (0.739 + 0.673i)25-s + (−0.739 + 0.673i)27-s + (−0.739 − 0.673i)28-s + (1.18 − 1.56i)31-s + (0.0922 − 0.995i)36-s + (0.365 + 1.95i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2163\)    =    \(3 \cdot 7 \cdot 103\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(1.07947\)
Root analytic conductor: \(1.03897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2163} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2163,\ (\ :0),\ 0.993 + 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373495308\)
\(L(\frac12)\) \(\approx\) \(1.373495308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.273 + 0.961i)T \)
7 \( 1 + (0.932 - 0.361i)T \)
103 \( 1 + (0.982 - 0.183i)T \)
good2 \( 1 + (-0.445 - 0.895i)T^{2} \)
5 \( 1 + (-0.739 - 0.673i)T^{2} \)
11 \( 1 + (0.445 + 0.895i)T^{2} \)
13 \( 1 + (-1.78 + 0.165i)T + (0.982 - 0.183i)T^{2} \)
17 \( 1 + (0.0922 + 0.995i)T^{2} \)
19 \( 1 + (-0.694 + 0.197i)T + (0.850 - 0.526i)T^{2} \)
23 \( 1 + (-0.445 + 0.895i)T^{2} \)
29 \( 1 + (-0.739 - 0.673i)T^{2} \)
31 \( 1 + (-1.18 + 1.56i)T + (-0.273 - 0.961i)T^{2} \)
37 \( 1 + (-0.365 - 1.95i)T + (-0.932 + 0.361i)T^{2} \)
41 \( 1 + (0.739 - 0.673i)T^{2} \)
43 \( 1 + (-0.247 + 1.32i)T + (-0.932 - 0.361i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.850 + 0.526i)T^{2} \)
59 \( 1 + (-0.982 - 0.183i)T^{2} \)
61 \( 1 + (1.07 - 1.17i)T + (-0.0922 - 0.995i)T^{2} \)
67 \( 1 + (-1.04 - 0.0971i)T + (0.982 + 0.183i)T^{2} \)
71 \( 1 + (0.739 - 0.673i)T^{2} \)
73 \( 1 + (1.12 - 0.435i)T + (0.739 - 0.673i)T^{2} \)
79 \( 1 + (0.172 + 0.0666i)T + (0.739 + 0.673i)T^{2} \)
83 \( 1 + (-0.982 + 0.183i)T^{2} \)
89 \( 1 + (0.602 + 0.798i)T^{2} \)
97 \( 1 + (0.907 + 0.995i)T + (-0.0922 + 0.995i)T^{2} \)
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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

−8.902724401767102773757995724802, −8.461671120589532723407951779976, −7.71079453938802898603939943377, −6.87455917731715014771540043509, −6.33239071423421460016579244381, −5.64626844390859461988490462589, −4.04281986411329741889959075343, −3.18259233062178868876097989956, −2.65753083982203982120686445302, −1.27583451935566295269408894582, 1.16706030675551327196820974294, 2.69652383574870129079961888939, 3.48267904115175474375462218822, 4.37408721595902459372746080969, 5.34797630401868409827623447889, 6.16834965915196711328178400602, 6.64542606986213907537606551133, 7.81345377500446127028233755391, 8.785856330396299622988232481947, 9.347673693734299413440931408584

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