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Properties

Label 2-2156-7.4-c1-0-13
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  + (0.707 − 1.22i)3-s + (0.500 + 0.866i)9-s + (0.5 − 0.866i)11-s + 1.41·13-s + (−3.53 + 6.12i)17-s + (1.41 + 2.44i)19-s + (2 + 3.46i)23-s + (2.5 − 4.33i)25-s + 5.65·27-s + 2·29-s + (−2.12 + 3.67i)31-s + (−0.707 − 1.22i)33-s + (2 + 3.46i)37-s + (1.00 − 1.73i)39-s − 1.41·41-s + ⋯
L(s)  = 1  + (0.408 − 0.707i)3-s + (0.166 + 0.288i)9-s + (0.150 − 0.261i)11-s + 0.392·13-s + (−0.857 + 1.48i)17-s + (0.324 + 0.561i)19-s + (0.417 + 0.722i)23-s + (0.5 − 0.866i)25-s + 1.08·27-s + 0.371·29-s + (−0.381 + 0.659i)31-s + (−0.123 − 0.213i)33-s + (0.328 + 0.569i)37-s + (0.160 − 0.277i)39-s − 0.220·41-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

\(L(1)\) \(\approx\) \(2.117546590\)
\(L(\frac12)\) \(\approx\) \(2.117546590\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + (3.53 - 6.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (2.12 - 3.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-4.94 - 8.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.12 + 3.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.36 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.48T + 83T^{2} \)
89 \( 1 + (-5.65 - 9.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.48T + 97T^{2} \)
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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

−8.846512966559789157660729230671, −8.271412877891261898249201306546, −7.68306485728899263419824817733, −6.69999265539355252856615103271, −6.19913271860185442274621953720, −5.10263312365293997878856193351, −4.13894262629125312046146407048, −3.18501656123145836384266699957, −2.07962606377655548064038089994, −1.20921582178916836626915820982, 0.810889429789893705305067641421, 2.41499464965090300152443686101, 3.24532167763463205435077415999, 4.24742316397698469615886274129, 4.82547367399436220357001523186, 5.83802496510041860472718140662, 6.96860134462120088880204811471, 7.28852330944209207686613053924, 8.704789028196061167402114690447, 8.992951849696607716599342185315

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