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Properties

Label 2-276-69.68-c1-0-2
Degree $2$
Conductor $276$
Sign $0.281 - 0.959i$
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  + (−0.780 + 1.54i)3-s + 3.02·5-s + 2.62i·7-s + (−1.78 − 2.41i)9-s + 1.32·11-s − 1.56·13-s + (−2.35 + 4.66i)15-s + 0.371·17-s + 6.71i·19-s + (−4.05 − 2.04i)21-s + (4.71 − 0.868i)23-s + 4.12·25-s + (5.12 − 0.868i)27-s + 4.82i·29-s − 8.68·31-s + ⋯
L(s)  = 1  + (−0.450 + 0.892i)3-s + 1.35·5-s + 0.991i·7-s + (−0.593 − 0.804i)9-s + 0.399·11-s − 0.433·13-s + (−0.608 + 1.20i)15-s + 0.0901·17-s + 1.54i·19-s + (−0.884 − 0.446i)21-s + (0.983 − 0.181i)23-s + 0.824·25-s + (0.985 − 0.167i)27-s + 0.896i·29-s − 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05702 + 0.791278i\)
\(L(\frac12)\) \(\approx\) \(1.05702 + 0.791278i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.780 - 1.54i)T \)
23 \( 1 + (-4.71 + 0.868i)T \)
good5 \( 1 - 3.02T + 5T^{2} \)
7 \( 1 - 2.62iT - 7T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 + 1.56T + 13T^{2} \)
17 \( 1 - 0.371T + 17T^{2} \)
19 \( 1 - 6.71iT - 19T^{2} \)
29 \( 1 - 4.82iT - 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 + 9.33iT - 37T^{2} \)
41 \( 1 + 1.35iT - 41T^{2} \)
43 \( 1 + 6.71iT - 43T^{2} \)
47 \( 1 + 11.0iT - 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 6.18iT - 59T^{2} \)
61 \( 1 + 5.24iT - 61T^{2} \)
67 \( 1 + 7.86iT - 67T^{2} \)
71 \( 1 - 3.09iT - 71T^{2} \)
73 \( 1 - 6.68T + 73T^{2} \)
79 \( 1 - 11.9iT - 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 7.73T + 89T^{2} \)
97 \( 1 - 4.09iT - 97T^{2} \)
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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

−12.12498310556016200257086696270, −10.95141736600775420785953948294, −10.12208750511703885134258092545, −9.319203593360877883324379894382, −8.711592867164575033280753521669, −6.88516718288997469525017068385, −5.60365682965563796483823565130, −5.41283402004393356135471671045, −3.66810252290702906778283339261, −2.09300117862288481132216940129, 1.20710062859298750325127459376, 2.65496413717944355497870004473, 4.69415630527137592386083491106, 5.79664580848538723618911885315, 6.77492718202045248225276753334, 7.45711199499895190703572143300, 8.914460266432002194017872790834, 9.855882555966938340577455243665, 10.83174469971407987061924744550, 11.62192530870736723734420832310

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