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Properties

Label 2-2156-7.2-c1-0-22
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  + (−1.69 − 2.92i)3-s + (−0.802 + 1.38i)5-s + (−4.21 + 7.29i)9-s + (−0.5 − 0.866i)11-s + 4.98·13-s + 5.42·15-s + (−0.887 − 1.53i)17-s + (3.38 − 5.85i)19-s + (0.712 − 1.23i)23-s + (1.21 + 2.09i)25-s + 18.3·27-s − 6·29-s + (−1.69 − 2.92i)31-s + (−1.69 + 2.92i)33-s + (2.71 − 4.69i)37-s + ⋯
L(s)  = 1  + (−0.975 − 1.69i)3-s + (−0.358 + 0.621i)5-s + (−1.40 + 2.43i)9-s + (−0.150 − 0.261i)11-s + 1.38·13-s + 1.40·15-s + (−0.215 − 0.372i)17-s + (0.775 − 1.34i)19-s + (0.148 − 0.257i)23-s + (0.242 + 0.419i)25-s + 3.52·27-s − 1.11·29-s + (−0.303 − 0.525i)31-s + (−0.294 + 0.509i)33-s + (0.445 − 0.772i)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

\(L(1)\) \(\approx\) \(0.5928937598\)
\(L(\frac12)\) \(\approx\) \(0.5928937598\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 2.92i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.802 - 1.38i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 4.98T + 13T^{2} \)
17 \( 1 + (0.887 + 1.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.38 + 5.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.712 + 1.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (1.69 + 2.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.71 + 4.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.19T + 41T^{2} \)
43 \( 1 + 8.84T + 43T^{2} \)
47 \( 1 + (-0.887 + 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.42 - 9.39i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0851 - 0.147i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.87 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.71 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.42T + 71T^{2} \)
73 \( 1 + (2.66 + 4.61i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.55T + 83T^{2} \)
89 \( 1 + (5.95 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.36T + 97T^{2} \)
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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

−8.406987831761687949182834150246, −7.64603895900867744198504231595, −7.01345972429314121959120108604, −6.52032632218503913545460015437, −5.68124745200946318477471270223, −5.00361511856524881955794531849, −3.50393299776644175009464825885, −2.50638317841795320667832620720, −1.38315634988011351028037410773, −0.27238093866262987469409157826, 1.26385374939106260607945373442, 3.36435366774695125743136028035, 3.85263009304727524059645027533, 4.65852960113267510906891423530, 5.46707040316620964567738307031, 5.95534524494359188468673466038, 6.95525532307627157089806921230, 8.438323152586806221885078080996, 8.622194733472543745665891614956, 9.756287119092866556126845409102

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