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Properties

Label 2-276-4.3-c2-0-10
Degree $2$
Conductor $276$
Sign $0.997 - 0.0758i$
Analytic cond. $7.52045$
Root an. cond. $2.74234$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0758 − 1.99i)2-s − 1.73i·3-s + (−3.98 + 0.303i)4-s + 1.44·5-s + (−3.46 + 0.131i)6-s + 10.3i·7-s + (0.908 + 7.94i)8-s − 2.99·9-s + (−0.109 − 2.88i)10-s + 19.0i·11-s + (0.525 + 6.90i)12-s + 20.2·13-s + (20.7 − 0.788i)14-s − 2.50i·15-s + (15.8 − 2.41i)16-s − 27.6·17-s + ⋯
L(s)  = 1  + (−0.0379 − 0.999i)2-s − 0.577i·3-s + (−0.997 + 0.0758i)4-s + 0.289·5-s + (−0.576 + 0.0219i)6-s + 1.48i·7-s + (0.113 + 0.993i)8-s − 0.333·9-s + (−0.0109 − 0.288i)10-s + 1.72i·11-s + (0.0437 + 0.575i)12-s + 1.55·13-s + (1.48 − 0.0562i)14-s − 0.166i·15-s + (0.988 − 0.151i)16-s − 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(7.52045\)
Root analytic conductor: \(2.74234\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1),\ 0.997 - 0.0758i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.25359 + 0.0475976i\)
\(L(\frac12)\) \(\approx\) \(1.25359 + 0.0475976i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0758 + 1.99i)T \)
3 \( 1 + 1.73iT \)
23 \( 1 - 4.79iT \)
good5 \( 1 - 1.44T + 25T^{2} \)
7 \( 1 - 10.3iT - 49T^{2} \)
11 \( 1 - 19.0iT - 121T^{2} \)
13 \( 1 - 20.2T + 169T^{2} \)
17 \( 1 + 27.6T + 289T^{2} \)
19 \( 1 + 5.26iT - 361T^{2} \)
29 \( 1 - 1.54T + 841T^{2} \)
31 \( 1 - 27.7iT - 961T^{2} \)
37 \( 1 - 42.8T + 1.36e3T^{2} \)
41 \( 1 - 36.9T + 1.68e3T^{2} \)
43 \( 1 - 6.35iT - 1.84e3T^{2} \)
47 \( 1 - 0.999iT - 2.20e3T^{2} \)
53 \( 1 + 36.9T + 2.80e3T^{2} \)
59 \( 1 + 41.4iT - 3.48e3T^{2} \)
61 \( 1 - 49.1T + 3.72e3T^{2} \)
67 \( 1 - 85.3iT - 4.48e3T^{2} \)
71 \( 1 + 5.34iT - 5.04e3T^{2} \)
73 \( 1 - 100.T + 5.32e3T^{2} \)
79 \( 1 - 136. iT - 6.24e3T^{2} \)
83 \( 1 + 58.3iT - 6.88e3T^{2} \)
89 \( 1 + 76.4T + 7.92e3T^{2} \)
97 \( 1 - 47.7T + 9.40e3T^{2} \)
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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

−11.66751590043549793993913427051, −11.01450449310566578401562161060, −9.664852329436885179740764970752, −8.995849319781514491683871937327, −8.141876657692577182549152848567, −6.60519698499719633468154138239, −5.50990728644457249569108302873, −4.26721740533576664268177139132, −2.55845245071763579777098699273, −1.73472631120725252853871131501, 0.67210465856230523153202813964, 3.63658516179686787471045554139, 4.32401505412488096844996044187, 5.87912720611337994873328160548, 6.46343041091606989164921880729, 7.86217943216768399504898499939, 8.642772130075186755992227281140, 9.572357397060527007004924487787, 10.77645975890434060520740771076, 11.15820174831469364166265044067

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