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Properties

Label 2-2156-44.27-c0-0-0
Degree $2$
Conductor $2156$
Sign $-0.550 - 0.835i$
Analytic cond. $1.07598$
Root an. cond. $1.03729$
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  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.309 + 0.951i)16-s − 0.999·18-s + (−0.809 − 0.587i)22-s − 1.17i·23-s + (0.809 + 0.587i)25-s + (0.5 − 0.363i)29-s − 32-s + (0.309 − 0.951i)36-s + (−1.30 + 0.951i)37-s + 1.90i·43-s + (0.809 − 0.587i)44-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.309 + 0.951i)16-s − 0.999·18-s + (−0.809 − 0.587i)22-s − 1.17i·23-s + (0.809 + 0.587i)25-s + (0.5 − 0.363i)29-s − 32-s + (0.309 − 0.951i)36-s + (−1.30 + 0.951i)37-s + 1.90i·43-s + (0.809 − 0.587i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.550 - 0.835i$
Analytic conductor: \(1.07598\)
Root analytic conductor: \(1.03729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :0),\ -0.550 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8580730275\)
\(L(\frac12)\) \(\approx\) \(0.8580730275\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 \)
11 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.17iT - T^{2} \)
29 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - 1.90iT - T^{2} \)
71 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.467505026928438029702706697609, −8.506552425724626378548684345989, −8.018345423731126531092063474979, −7.12591800608714391228299774248, −6.67729098221780197741524123845, −5.56205298037299800198568058221, −4.80990440759640955582840321610, −4.27812851955348525335795527446, −2.72340300795990154534901209859, −1.46071203357738568918175189611, 0.72948964842584165219139244900, 2.00892457246807112872491407194, 3.27538017577820320281930120356, 3.68779421886326975714011881879, 4.86301620328749285179896585610, 5.69859108863680616510472584177, 6.77781764619308280782987155638, 7.59164252285173283942518069785, 8.598775670412062856257815046948, 8.961099125997195414237449466080

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