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Properties

Label 2-276-23.18-c1-0-3
Degree $2$
Conductor $276$
Sign $0.118 + 0.992i$
Analytic cond. $2.20387$
Root an. cond. $1.48454$
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  + (0.654 − 0.755i)3-s + (0.355 − 2.47i)5-s + (−0.671 − 1.46i)7-s + (−0.142 − 0.989i)9-s + (−2.90 + 0.854i)11-s + (0.814 − 1.78i)13-s + (−1.63 − 1.89i)15-s + (−0.513 + 0.330i)17-s + (4.77 + 3.07i)19-s + (−1.55 − 0.455i)21-s + (−1.13 − 4.65i)23-s + (−1.20 − 0.354i)25-s + (−0.841 − 0.540i)27-s + (5.08 − 3.27i)29-s + (6.59 + 7.61i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (0.159 − 1.10i)5-s + (−0.253 − 0.555i)7-s + (−0.0474 − 0.329i)9-s + (−0.876 + 0.257i)11-s + (0.225 − 0.494i)13-s + (−0.422 − 0.488i)15-s + (−0.124 + 0.0800i)17-s + (1.09 + 0.704i)19-s + (−0.338 − 0.0993i)21-s + (−0.236 − 0.971i)23-s + (−0.241 − 0.0708i)25-s + (−0.161 − 0.104i)27-s + (0.945 − 0.607i)29-s + (1.18 + 1.36i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.118 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01756 - 0.903097i\)
\(L(\frac12)\) \(\approx\) \(1.01756 - 0.903097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (1.13 + 4.65i)T \)
good5 \( 1 + (-0.355 + 2.47i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.671 + 1.46i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (2.90 - 0.854i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.814 + 1.78i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.513 - 0.330i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-4.77 - 3.07i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-5.08 + 3.27i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-6.59 - 7.61i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.255 - 1.77i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (-0.781 + 5.43i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (6.13 - 7.07i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 1.60T + 47T^{2} \)
53 \( 1 + (-3.13 - 6.87i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (3.62 - 7.94i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-1.65 - 1.91i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-0.507 - 0.149i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (0.986 + 0.289i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-2.40 - 1.54i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (1.69 - 3.71i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-1.25 - 8.71i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (9.26 - 10.6i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-2.48 + 17.2i)T + (-93.0 - 27.3i)T^{2} \)
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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

−12.03074680481054593106418206526, −10.50979822344763505752230987815, −9.812280578770979538015850663089, −8.568165192636261884558462710803, −7.991769624994753160450921268655, −6.81556611499440064789738401256, −5.52526880237241238901395290869, −4.44529313668025280260626595673, −2.90210154867008648268873682966, −1.09235806862157605950108828204, 2.50201244381036368276217136234, 3.36779390974982555543932436327, 4.97204834794026077324350665290, 6.14859369117157590248861341755, 7.21509519556318115503325266886, 8.285948383653728588748816873509, 9.415305632099218700670346366162, 10.14853456814242277931167041753, 11.11117720481575759314484455911, 11.86373906795616235795829702567

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