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Properties

Label 2-2156-7.2-c1-0-30
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  + (1.42 + 2.47i)3-s + (2.07 − 3.59i)5-s + (−2.57 + 4.46i)9-s + (−0.5 − 0.866i)11-s + 5.29·13-s + 11.8·15-s + (−2.20 − 3.82i)17-s + (2.85 − 4.94i)19-s + (1.77 − 3.07i)23-s + (−6.13 − 10.6i)25-s − 6.15·27-s − 0.599·29-s + (−2.28 − 3.95i)31-s + (1.42 − 2.47i)33-s + (−3.07 + 5.33i)37-s + ⋯
L(s)  = 1  + (0.824 + 1.42i)3-s + (0.929 − 1.60i)5-s + (−0.859 + 1.48i)9-s + (−0.150 − 0.261i)11-s + 1.46·13-s + 3.06·15-s + (−0.535 − 0.926i)17-s + (0.655 − 1.13i)19-s + (0.370 − 0.642i)23-s + (−1.22 − 2.12i)25-s − 1.18·27-s − 0.111·29-s + (−0.410 − 0.710i)31-s + (0.248 − 0.430i)33-s + (−0.505 + 0.876i)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

\(L(1)\) \(\approx\) \(2.899475232\)
\(L(\frac12)\) \(\approx\) \(2.899475232\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 2.47i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.07 + 3.59i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 + (2.20 + 3.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.85 + 4.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.77 + 3.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.599T + 29T^{2} \)
31 \( 1 + (2.28 + 3.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.07 - 5.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.41T + 41T^{2} \)
43 \( 1 - 3.71T + 43T^{2} \)
47 \( 1 + (2.50 - 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.15 + 7.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.57 - 4.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.80 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.221 - 0.384i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.75T + 71T^{2} \)
73 \( 1 + (-4.20 - 7.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.01 + 6.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.82T + 83T^{2} \)
89 \( 1 + (3.78 - 6.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−9.085991570765355214015374797187, −8.695438517605737439193877431104, −7.979683473276241672417919654620, −6.54716056508219892131255206230, −5.53081887204234222444882207158, −4.92038518157549642241506774544, −4.34747080446529240375794798124, −3.33233577725733138356427528490, −2.32834271944809307286433956461, −0.955176925386372029789233842459, 1.58969737540673213346821962818, 1.98135973486348257669104159921, 3.20032941004876461154796352316, 3.59105355673818253511283064127, 5.58099791824100539291596663022, 6.24842609868796844741322557930, 6.75267450024696129819665672984, 7.47146242953313519105326531824, 8.141626542861256271336850365854, 8.981238488140113242549878917552

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