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Properties

Label 2-2163-2163.1007-c0-0-1
Degree $2$
Conductor $2163$
Sign $-0.700 - 0.713i$
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  + (0.850 + 0.526i)3-s + (−0.602 + 0.798i)4-s + (−0.739 + 0.673i)7-s + (0.445 + 0.895i)9-s + (−0.932 + 0.361i)12-s + (0.293 + 1.56i)13-s + (−0.273 − 0.961i)16-s + (−0.709 − 1.14i)19-s + (−0.982 + 0.183i)21-s + (0.0922 + 0.995i)25-s + (−0.0922 + 0.995i)27-s + (−0.0922 − 0.995i)28-s + (−0.510 − 1.79i)31-s + (−0.982 − 0.183i)36-s + (−0.132 − 0.342i)37-s + ⋯
L(s)  = 1  + (0.850 + 0.526i)3-s + (−0.602 + 0.798i)4-s + (−0.739 + 0.673i)7-s + (0.445 + 0.895i)9-s + (−0.932 + 0.361i)12-s + (0.293 + 1.56i)13-s + (−0.273 − 0.961i)16-s + (−0.709 − 1.14i)19-s + (−0.982 + 0.183i)21-s + (0.0922 + 0.995i)25-s + (−0.0922 + 0.995i)27-s + (−0.0922 − 0.995i)28-s + (−0.510 − 1.79i)31-s + (−0.982 − 0.183i)36-s + (−0.132 − 0.342i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2163\)    =    \(3 \cdot 7 \cdot 103\)
Sign: $-0.700 - 0.713i$
Analytic conductor: \(1.07947\)
Root analytic conductor: \(1.03897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2163} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2163,\ (\ :0),\ -0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102748564\)
\(L(\frac12)\) \(\approx\) \(1.102748564\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.850 - 0.526i)T \)
7 \( 1 + (0.739 - 0.673i)T \)
103 \( 1 + (-0.932 + 0.361i)T \)
good2 \( 1 + (0.602 - 0.798i)T^{2} \)
5 \( 1 + (-0.0922 - 0.995i)T^{2} \)
11 \( 1 + (-0.602 + 0.798i)T^{2} \)
13 \( 1 + (-0.293 - 1.56i)T + (-0.932 + 0.361i)T^{2} \)
17 \( 1 + (-0.982 + 0.183i)T^{2} \)
19 \( 1 + (0.709 + 1.14i)T + (-0.445 + 0.895i)T^{2} \)
23 \( 1 + (0.602 + 0.798i)T^{2} \)
29 \( 1 + (-0.0922 - 0.995i)T^{2} \)
31 \( 1 + (0.510 + 1.79i)T + (-0.850 + 0.526i)T^{2} \)
37 \( 1 + (0.132 + 0.342i)T + (-0.739 + 0.673i)T^{2} \)
41 \( 1 + (0.0922 - 0.995i)T^{2} \)
43 \( 1 + (0.719 - 1.85i)T + (-0.739 - 0.673i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.445 - 0.895i)T^{2} \)
59 \( 1 + (0.932 + 0.361i)T^{2} \)
61 \( 1 + (-1.91 + 0.177i)T + (0.982 - 0.183i)T^{2} \)
67 \( 1 + (0.328 - 1.75i)T + (-0.932 - 0.361i)T^{2} \)
71 \( 1 + (0.0922 - 0.995i)T^{2} \)
73 \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \)
79 \( 1 + (-1.45 - 1.32i)T + (0.0922 + 0.995i)T^{2} \)
83 \( 1 + (0.932 - 0.361i)T^{2} \)
89 \( 1 + (0.273 - 0.961i)T^{2} \)
97 \( 1 + (1.98 + 0.183i)T + (0.982 + 0.183i)T^{2} \)
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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

−9.473171491503323874246061746398, −8.827371552658988369713070624017, −8.313557293821094323475854424909, −7.29545706910425877308078517590, −6.63103439623813483839012918713, −5.41563238770494327057716136614, −4.41288977663302617499724149828, −3.88942354782150308045332843515, −2.93456296112042648656965240418, −2.12596405871658045183762756991, 0.70055597920992841013299559571, 1.90634387192981437354942386581, 3.27260342856206446107621688742, 3.81668478217814411019435017873, 4.97386121130613265445100310778, 5.97258454185083773677249077754, 6.59286288738530407074746476644, 7.48832330903394583538691908641, 8.403670078540262965140954547778, 8.761585563897696424029725530980

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