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Properties

Label 2-276-4.3-c2-0-43
Degree $2$
Conductor $276$
Sign $-0.330 - 0.943i$
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  + (1.15 − 1.63i)2-s − 1.73i·3-s + (−1.32 − 3.77i)4-s − 5.34·5-s + (−2.82 − 2.00i)6-s + 12.9i·7-s + (−7.68 − 2.21i)8-s − 2.99·9-s + (−6.18 + 8.71i)10-s − 7.61i·11-s + (−6.53 + 2.28i)12-s − 20.9·13-s + (21.2 + 15.0i)14-s + 9.25i·15-s + (−12.5 + 9.97i)16-s − 4.33·17-s + ⋯
L(s)  = 1  + (0.578 − 0.815i)2-s − 0.577i·3-s + (−0.330 − 0.943i)4-s − 1.06·5-s + (−0.470 − 0.334i)6-s + 1.85i·7-s + (−0.960 − 0.276i)8-s − 0.333·9-s + (−0.618 + 0.871i)10-s − 0.691i·11-s + (−0.544 + 0.190i)12-s − 1.60·13-s + (1.51 + 1.07i)14-s + 0.616i·15-s + (−0.781 + 0.623i)16-s − 0.254·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.330 - 0.943i$
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: \((2,\ 276,\ (\ :1),\ -0.330 - 0.943i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0706428 + 0.0995628i\)
\(L(\frac12)\) \(\approx\) \(0.0706428 + 0.0995628i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 1.63i)T \)
3 \( 1 + 1.73iT \)
23 \( 1 + 4.79iT \)
good5 \( 1 + 5.34T + 25T^{2} \)
7 \( 1 - 12.9iT - 49T^{2} \)
11 \( 1 + 7.61iT - 121T^{2} \)
13 \( 1 + 20.9T + 169T^{2} \)
17 \( 1 + 4.33T + 289T^{2} \)
19 \( 1 + 4.70iT - 361T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 - 11.3iT - 961T^{2} \)
37 \( 1 - 8.17T + 1.36e3T^{2} \)
41 \( 1 + 64.9T + 1.68e3T^{2} \)
43 \( 1 + 65.6iT - 1.84e3T^{2} \)
47 \( 1 + 62.7iT - 2.20e3T^{2} \)
53 \( 1 - 1.27T + 2.80e3T^{2} \)
59 \( 1 - 38.7iT - 3.48e3T^{2} \)
61 \( 1 + 80.1T + 3.72e3T^{2} \)
67 \( 1 + 12.1iT - 4.48e3T^{2} \)
71 \( 1 - 65.6iT - 5.04e3T^{2} \)
73 \( 1 + 89.4T + 5.32e3T^{2} \)
79 \( 1 - 127. iT - 6.24e3T^{2} \)
83 \( 1 + 125. iT - 6.88e3T^{2} \)
89 \( 1 - 34.7T + 7.92e3T^{2} \)
97 \( 1 - 142.T + 9.40e3T^{2} \)
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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

−11.57994649878457517100714340002, −10.25493512154546639485974166282, −8.996142497142212699659621179706, −8.334545248272329454584054885481, −6.89823705055945209959630841609, −5.67710242627587625794326587288, −4.77595343830089140242089215303, −3.15833715419397369831896196441, −2.23005333065100768292575298226, −0.04799665739992655757169509022, 3.24867035115860252721004772077, 4.41147456468141795905655429480, 4.70222074679734254402932280680, 6.60166288176956595153486442623, 7.51616843711508916581090208423, 7.941316062941316218263557227368, 9.517580973107445644098679208553, 10.38186111382207405526638219223, 11.54445362222804372982817117147, 12.34687463026389610468028043217

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