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Properties

Label 2-2156-7.4-c1-0-5
Degree $2$
Conductor $2156$
Sign $-0.991 + 0.126i$
Analytic cond. $17.2157$
Root an. cond. $4.14918$
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 + (1.5 + 2.59i)5-s + (1 + 1.73i)9-s + (0.5 − 0.866i)11-s − 4·13-s − 3·15-s + (−3 + 5.19i)17-s + (−4 − 6.92i)19-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s − 5·27-s + (−2.5 + 4.33i)31-s + (0.499 + 0.866i)33-s + (0.5 + 0.866i)37-s + (2 − 3.46i)39-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.670 + 1.16i)5-s + (0.333 + 0.577i)9-s + (0.150 − 0.261i)11-s − 1.10·13-s − 0.774·15-s + (−0.727 + 1.26i)17-s + (−0.917 − 1.58i)19-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.962·27-s + (−0.449 + 0.777i)31-s + (0.0870 + 0.150i)33-s + (0.0821 + 0.142i)37-s + (0.320 − 0.554i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (1145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9123831301\)
\(L(\frac12)\) \(\approx\) \(0.9123831301\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−9.646607645759054046137974914087, −8.839880730912391657347691186684, −7.88037807443422884732617997125, −6.79439039161101389638796102348, −6.63157028160009452573068518083, −5.42619355643597567989477200556, −4.76770504851687484346015379312, −3.77229161854014070862065872863, −2.64195737578491255764831793282, −1.92070140309929650413065459521, 0.30791476009999026734898523567, 1.53448885903603392648822590898, 2.38237176205914225597149531827, 3.91133395522680153929712870254, 4.77073721108431445045230278798, 5.44816726269419656633045030770, 6.36601718689926981296252211529, 7.00795988353557304507964240780, 7.905181756568216937940783618826, 8.747584011157596826577554754734

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