Properties

Label 2-276-23.16-c1-0-0
Degree $2$
Conductor $276$
Sign $-0.694 - 0.719i$
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  + (−0.841 + 0.540i)3-s + (1.63 + 3.58i)5-s + (−4.27 + 1.25i)7-s + (0.415 − 0.909i)9-s + (−3.34 − 3.85i)11-s + (−1.02 − 0.299i)13-s + (−3.31 − 2.13i)15-s + (−0.521 + 3.62i)17-s + (0.785 + 5.46i)19-s + (2.91 − 3.36i)21-s + (4.78 + 0.369i)23-s + (−6.92 + 7.98i)25-s + (0.142 + 0.989i)27-s + (0.455 − 3.17i)29-s + (8.45 + 5.43i)31-s + ⋯
L(s)  = 1  + (−0.485 + 0.312i)3-s + (0.733 + 1.60i)5-s + (−1.61 + 0.474i)7-s + (0.138 − 0.303i)9-s + (−1.00 − 1.16i)11-s + (−0.283 − 0.0831i)13-s + (−0.857 − 0.550i)15-s + (−0.126 + 0.879i)17-s + (0.180 + 1.25i)19-s + (0.636 − 0.734i)21-s + (0.997 + 0.0769i)23-s + (−1.38 + 1.59i)25-s + (0.0273 + 0.190i)27-s + (0.0846 − 0.588i)29-s + (1.51 + 0.975i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.694 - 0.719i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304495 + 0.717347i\)
\(L(\frac12)\) \(\approx\) \(0.304495 + 0.717347i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-4.78 - 0.369i)T \)
good5 \( 1 + (-1.63 - 3.58i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (4.27 - 1.25i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (3.34 + 3.85i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (1.02 + 0.299i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.521 - 3.62i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.785 - 5.46i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.455 + 3.17i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-8.45 - 5.43i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (1.68 - 3.68i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (1.54 + 3.38i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (4.37 - 2.81i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 4.77T + 47T^{2} \)
53 \( 1 + (-3.17 + 0.933i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-5.48 - 1.61i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-0.386 - 0.248i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (2.29 - 2.65i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-2.72 + 3.13i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.791 - 5.50i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (6.23 + 1.82i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (3.57 - 7.82i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-3.64 + 2.34i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-2.34 - 5.12i)T + (-63.5 + 73.3i)T^{2} \)
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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

−12.20344224186336222620069254331, −11.02936092952319416401762170788, −10.19844801194662227066158360456, −9.924105801814333317151894276756, −8.477793041337245596622986526794, −6.96517897610815455251783763700, −6.18119256922201925161108827751, −5.60965534094588823349540859873, −3.42978560382071789034289884824, −2.75215750082820782407568306670, 0.60383162427117390089909374458, 2.55785122654590644699570523713, 4.60968440789914056012115325840, 5.28519140227101708208198204753, 6.57437841333556867273742830353, 7.42325668893438807145676157750, 8.931909049030843574421316148686, 9.653235889596830860718428335408, 10.30578197171612739625671104677, 11.82773807047035165491744637055

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