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Properties

Label 2-2156-44.31-c0-0-1
Degree $2$
Conductor $2156$
Sign $0.0457 + 0.998i$
Analytic cond. $1.07598$
Root an. cond. $1.03729$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.809 + 0.587i)22-s − 1.17i·23-s + (0.809 − 0.587i)25-s + (0.5 + 0.363i)29-s − 32-s + (−0.809 − 0.587i)36-s + (−1.30 − 0.951i)37-s + 1.90i·43-s + 44-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.809 + 0.587i)22-s − 1.17i·23-s + (0.809 − 0.587i)25-s + (0.5 + 0.363i)29-s − 32-s + (−0.809 − 0.587i)36-s + (−1.30 − 0.951i)37-s + 1.90i·43-s + 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.0457 + 0.998i$
Analytic conductor: \(1.07598\)
Root analytic conductor: \(1.03729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :0),\ 0.0457 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.882926496\)
\(L(\frac12)\) \(\approx\) \(1.882926496\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 \)
11 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.17iT - T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - 1.90iT - T^{2} \)
71 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−9.260619221646573952632310258993, −8.495189521230951502256498580448, −7.12300891015867356796205536148, −6.70132575449170887587267911925, −5.88619258511561184115702576009, −4.78564509680483174264859772263, −4.25746221996550414727774721627, −3.29622688538725096910354894141, −2.32198143447043050283840273335, −1.12812718897461128216231276572, 1.77272992327997460524891031736, 3.02559129580826860242237050292, 3.75375999522768963027245614725, 4.83291672661931920139357735444, 5.39930400202314447065443570226, 6.25604237199220609150750650274, 7.09382114724069700988116585290, 7.74241093801919321006829826139, 8.526718897051506495801929374660, 9.170270498135628545332788761732

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