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Properties

Label 2-276-69.65-c1-0-3
Degree $2$
Conductor $276$
Sign $0.876 - 0.481i$
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  + (1.72 + 0.128i)3-s + (2.86 + 1.84i)5-s + (−2.95 + 0.425i)7-s + (2.96 + 0.443i)9-s + (−1.62 + 3.55i)11-s + (0.542 − 3.77i)13-s + (4.71 + 3.54i)15-s + (−2.99 − 3.45i)17-s + (−1.65 − 1.43i)19-s + (−5.16 + 0.354i)21-s + (−3.59 − 3.16i)23-s + (2.73 + 5.99i)25-s + (5.06 + 1.14i)27-s + (6.23 − 5.40i)29-s + (10.2 + 3.01i)31-s + ⋯
L(s)  = 1  + (0.997 + 0.0741i)3-s + (1.28 + 0.823i)5-s + (−1.11 + 0.160i)7-s + (0.989 + 0.147i)9-s + (−0.489 + 1.07i)11-s + (0.150 − 1.04i)13-s + (1.21 + 0.915i)15-s + (−0.725 − 0.837i)17-s + (−0.380 − 0.329i)19-s + (−1.12 + 0.0774i)21-s + (−0.750 − 0.660i)23-s + (0.547 + 1.19i)25-s + (0.975 + 0.220i)27-s + (1.15 − 1.00i)29-s + (1.84 + 0.541i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79481 + 0.460636i\)
\(L(\frac12)\) \(\approx\) \(1.79481 + 0.460636i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.128i)T \)
23 \( 1 + (3.59 + 3.16i)T \)
good5 \( 1 + (-2.86 - 1.84i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (2.95 - 0.425i)T + (6.71 - 1.97i)T^{2} \)
11 \( 1 + (1.62 - 3.55i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.542 + 3.77i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (2.99 + 3.45i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (1.65 + 1.43i)T + (2.70 + 18.8i)T^{2} \)
29 \( 1 + (-6.23 + 5.40i)T + (4.12 - 28.7i)T^{2} \)
31 \( 1 + (-10.2 - 3.01i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (3.75 + 5.84i)T + (-15.3 + 33.6i)T^{2} \)
41 \( 1 + (4.45 - 6.93i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-0.0854 - 0.291i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + 1.92iT - 47T^{2} \)
53 \( 1 + (-0.0778 - 0.541i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (9.74 + 1.40i)T + (56.6 + 16.6i)T^{2} \)
61 \( 1 + (-0.844 + 2.87i)T + (-51.3 - 32.9i)T^{2} \)
67 \( 1 + (-0.864 + 0.394i)T + (43.8 - 50.6i)T^{2} \)
71 \( 1 + (10.6 - 4.84i)T + (46.4 - 53.6i)T^{2} \)
73 \( 1 + (0.782 - 0.902i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (2.71 + 0.390i)T + (75.7 + 22.2i)T^{2} \)
83 \( 1 + (1.72 - 1.10i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-2.53 + 0.743i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (7.53 - 11.7i)T + (-40.2 - 88.2i)T^{2} \)
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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

−12.25165145952436239208501936899, −10.40711370971044065546399224849, −10.09475066466647631240318204190, −9.353287000032143374931278279970, −8.165266892233794024178122211223, −6.90552142028983480108460369535, −6.24788724224525219147422595201, −4.68608857742373642286692434664, −2.94782417810701065319535360588, −2.37754845881518914897306124570, 1.68932206937890345561217898772, 3.07614609163127473088401702380, 4.43105338423472428683410353902, 5.98100724519851280757807230750, 6.68770665745692268162715300907, 8.359323660619891568483982059362, 8.880561310544993445851628472763, 9.816074101868910571639283857984, 10.43806757248525311107081549452, 12.12717763555063332702203190405

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