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Properties

Label 2-276-92.91-c1-0-17
Degree $2$
Conductor $276$
Sign $0.971 + 0.237i$
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  + (1.40 − 0.174i)2-s + i·3-s + (1.93 − 0.489i)4-s − 2.63i·5-s + (0.174 + 1.40i)6-s − 0.526·7-s + (2.63 − 1.02i)8-s − 9-s + (−0.459 − 3.69i)10-s + 4.16·11-s + (0.489 + 1.93i)12-s − 1.45·13-s + (−0.738 + 0.0918i)14-s + 2.63·15-s + (3.52 − 1.89i)16-s + 7.00i·17-s + ⋯
L(s)  = 1  + (0.992 − 0.123i)2-s + 0.577i·3-s + (0.969 − 0.244i)4-s − 1.17i·5-s + (0.0712 + 0.572i)6-s − 0.198·7-s + (0.931 − 0.362i)8-s − 0.333·9-s + (−0.145 − 1.16i)10-s + 1.25·11-s + (0.141 + 0.559i)12-s − 0.403·13-s + (−0.197 + 0.0245i)14-s + 0.680·15-s + (0.880 − 0.474i)16-s + 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.971 + 0.237i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24676 - 0.270099i\)
\(L(\frac12)\) \(\approx\) \(2.24676 - 0.270099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.174i)T \)
3 \( 1 - iT \)
23 \( 1 + (2.24 - 4.23i)T \)
good5 \( 1 + 2.63iT - 5T^{2} \)
7 \( 1 + 0.526T + 7T^{2} \)
11 \( 1 - 4.16T + 11T^{2} \)
13 \( 1 + 1.45T + 13T^{2} \)
17 \( 1 - 7.00iT - 17T^{2} \)
19 \( 1 + 7.45T + 19T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + 8.01iT - 31T^{2} \)
37 \( 1 - 2.39iT - 37T^{2} \)
41 \( 1 + 4.81T + 41T^{2} \)
43 \( 1 - 8.10T + 43T^{2} \)
47 \( 1 + 1.90iT - 47T^{2} \)
53 \( 1 + 6.35iT - 53T^{2} \)
59 \( 1 - 7.78iT - 59T^{2} \)
61 \( 1 + 0.558iT - 61T^{2} \)
67 \( 1 + 7.34T + 67T^{2} \)
71 \( 1 - 8.61iT - 71T^{2} \)
73 \( 1 - 0.375T + 73T^{2} \)
79 \( 1 + 0.238T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 - 9.60iT - 89T^{2} \)
97 \( 1 - 3.43iT - 97T^{2} \)
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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

−12.04200991067185071777110441465, −11.10643906020078352596431907671, −10.05757621420644482154250835931, −9.068379065653257541652909124548, −8.051239373589724217904154618908, −6.47785670566429107015178697574, −5.65270872639253509668139286302, −4.36484532986788351931539996631, −3.84183145315258650661656186108, −1.81295907926523132908368528950, 2.22993497343052322482260095142, 3.32915077602312490955829996277, 4.66429168555690157193011590198, 6.24052411162765138237876292622, 6.73988654633824311685786388386, 7.52981048026274730771853821131, 8.985668367778562033509109374574, 10.40904921075305133822422559392, 11.21309882575696986553872929946, 12.05475131447386602481536175841

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