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Properties

Label 2-2156-7.4-c1-0-21
Degree $2$
Conductor $2156$
Sign $0.266 + 0.963i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.266 + 0.963i$
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: \((2,\ 2156,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.999357953\)
\(L(\frac12)\) \(\approx\) \(1.999357953\)
\(L(\frac{3}{2})\) 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

−8.705393728190195645129712746234, −8.027351960841966525162165316499, −7.70730810420885812803516143315, −6.69552548130178812988913594951, −5.58204974442550100257649658514, −4.98093259047788555916281024549, −3.95271084631008657067094350898, −3.16459147765495697553038880501, −1.65110855444307925078771647626, −0.867019606705551670569997634295, 1.15719486981385096134488514759, 2.80633193980342424731671045718, 3.47842469105465182038166286261, 4.08471869641801425897550435525, 5.15490148434485990865671274825, 6.40854835744296502183307116274, 6.79198424547580036031874699422, 7.66496033398722562270709018524, 8.541581226484466838900517859518, 9.180842815514834691994655601758

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