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Properties

Label 2-276-4.3-c2-0-14
Degree $2$
Conductor $276$
Sign $0.856 - 0.515i$
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  + (−1.92 + 0.535i)2-s + 1.73i·3-s + (3.42 − 2.06i)4-s − 6.57·5-s + (−0.927 − 3.33i)6-s − 5.60i·7-s + (−5.49 + 5.81i)8-s − 2.99·9-s + (12.6 − 3.52i)10-s + 0.677i·11-s + (3.57 + 5.93i)12-s + 13.4·13-s + (3.00 + 10.7i)14-s − 11.3i·15-s + (7.48 − 14.1i)16-s + 14.4·17-s + ⋯
L(s)  = 1  + (−0.963 + 0.267i)2-s + 0.577i·3-s + (0.856 − 0.515i)4-s − 1.31·5-s + (−0.154 − 0.556i)6-s − 0.800i·7-s + (−0.687 + 0.726i)8-s − 0.333·9-s + (1.26 − 0.352i)10-s + 0.0615i·11-s + (0.297 + 0.494i)12-s + 1.03·13-s + (0.214 + 0.771i)14-s − 0.759i·15-s + (0.467 − 0.883i)16-s + 0.851·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.856 - 0.515i$
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: \((2,\ 276,\ (\ :1),\ 0.856 - 0.515i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.774279 + 0.215180i\)
\(L(\frac12)\) \(\approx\) \(0.774279 + 0.215180i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.92 - 0.535i)T \)
3 \( 1 - 1.73iT \)
23 \( 1 - 4.79iT \)
good5 \( 1 + 6.57T + 25T^{2} \)
7 \( 1 + 5.60iT - 49T^{2} \)
11 \( 1 - 0.677iT - 121T^{2} \)
13 \( 1 - 13.4T + 169T^{2} \)
17 \( 1 - 14.4T + 289T^{2} \)
19 \( 1 - 4.32iT - 361T^{2} \)
29 \( 1 - 28.9T + 841T^{2} \)
31 \( 1 - 17.7iT - 961T^{2} \)
37 \( 1 - 41.6T + 1.36e3T^{2} \)
41 \( 1 - 36.9T + 1.68e3T^{2} \)
43 \( 1 - 32.3iT - 1.84e3T^{2} \)
47 \( 1 + 40.4iT - 2.20e3T^{2} \)
53 \( 1 - 34.0T + 2.80e3T^{2} \)
59 \( 1 - 110. iT - 3.48e3T^{2} \)
61 \( 1 - 39.3T + 3.72e3T^{2} \)
67 \( 1 + 34.0iT - 4.48e3T^{2} \)
71 \( 1 + 106. iT - 5.04e3T^{2} \)
73 \( 1 - 25.5T + 5.32e3T^{2} \)
79 \( 1 - 9.81iT - 6.24e3T^{2} \)
83 \( 1 + 135. iT - 6.88e3T^{2} \)
89 \( 1 + 121.T + 7.92e3T^{2} \)
97 \( 1 + 137.T + 9.40e3T^{2} \)
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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

−11.42099905996660773362548824067, −10.71033569040302105780868492944, −9.921141199649680794671107267441, −8.721942276506076036386896386172, −7.961879403757053732611463467511, −7.16936642283787478600119869356, −5.89955061713398329948149146254, −4.34751396414665854876322201680, −3.25893724994025810214509453900, −0.879960037126609907447801389423, 0.864966189320814552252235719776, 2.65454726762119392870135658408, 3.88773588705751457990872729293, 5.84203191886405246507005132458, 6.94818134477717161408690190863, 8.018267093272049464890613837115, 8.422872870682749020930635192954, 9.525856031513045007582115505896, 10.86678523385837235512442212926, 11.55721288713602621849872095559

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