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Properties

Label 2-2156-7.2-c1-0-11
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  + (−1.08 − 1.88i)3-s + (0.370 − 0.641i)5-s + (−0.870 + 1.50i)9-s + (−0.5 − 0.866i)11-s + 6.91·13-s − 1.61·15-s + (3.63 + 6.29i)17-s + (−2.17 + 3.77i)19-s + (−1.54 + 2.68i)23-s + (2.22 + 3.85i)25-s − 2.74·27-s − 3.83·29-s + (5.26 + 9.12i)31-s + (−1.08 + 1.88i)33-s + (−1.37 + 2.37i)37-s + ⋯
L(s)  = 1  + (−0.628 − 1.08i)3-s + (0.165 − 0.287i)5-s + (−0.290 + 0.502i)9-s + (−0.150 − 0.261i)11-s + 1.91·13-s − 0.416·15-s + (0.882 + 1.52i)17-s + (−0.499 + 0.865i)19-s + (−0.322 + 0.559i)23-s + (0.445 + 0.770i)25-s − 0.527·27-s − 0.712·29-s + (0.945 + 1.63i)31-s + (−0.189 + 0.328i)33-s + (−0.225 + 0.390i)37-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: $\chi_{2156} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.479878843\)
\(L(\frac12)\) \(\approx\) \(1.479878843\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 + 1.88i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.370 + 0.641i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 6.91T + 13T^{2} \)
17 \( 1 + (-3.63 - 6.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.17 - 3.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.54 - 2.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.83T + 29T^{2} \)
31 \( 1 + (-5.26 - 9.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.37 - 2.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.27T + 41T^{2} \)
43 \( 1 + 6.35T + 43T^{2} \)
47 \( 1 + (-1.71 + 2.97i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.741 + 1.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.08 - 8.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.20 - 7.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.54 - 6.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.57T + 71T^{2} \)
73 \( 1 + (1.63 + 2.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.43 - 7.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (-7.98 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.386T + 97T^{2} \)
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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

−8.659722195313041728418222815523, −8.391325577068825146881463402661, −7.46217044797261192713179576159, −6.58029613472264027634703214958, −5.88059289504585532881348219409, −5.56850253175645580746349895060, −4.05221867287954568170091860159, −3.31886685067085957193772723254, −1.57473901869721133318215842245, −1.24113515488755138255938071315, 0.65650706905593981541892281573, 2.35937476240071002262227750999, 3.46618156841522113953503379016, 4.31414699975348663112061228343, 5.01521522007736959006336987975, 5.91876810150712411116053880891, 6.48579518293783010287790128650, 7.57063935593827646960290114811, 8.421997223541067348121328188385, 9.339876321633734605146642411700

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