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Properties

Label 2-276-92.91-c1-0-8
Degree $2$
Conductor $276$
Sign $0.663 + 0.748i$
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  + (−0.588 − 1.28i)2-s − i·3-s + (−1.30 + 1.51i)4-s + 2.28i·5-s + (−1.28 + 0.588i)6-s + 3.41·7-s + (2.71 + 0.792i)8-s − 9-s + (2.94 − 1.34i)10-s + 1.67·11-s + (1.51 + 1.30i)12-s + 0.215·13-s + (−2.00 − 4.39i)14-s + 2.28·15-s + (−0.577 − 3.95i)16-s + 0.222i·17-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s − 0.577i·3-s + (−0.654 + 0.756i)4-s + 1.02i·5-s + (−0.525 + 0.240i)6-s + 1.29·7-s + (0.959 + 0.280i)8-s − 0.333·9-s + (0.930 − 0.425i)10-s + 0.503·11-s + (0.436 + 0.377i)12-s + 0.0598·13-s + (−0.536 − 1.17i)14-s + 0.590·15-s + (−0.144 − 0.989i)16-s + 0.0540i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.663 + 0.748i$
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: \((2,\ 276,\ (\ :1/2),\ 0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03334 - 0.464867i\)
\(L(\frac12)\) \(\approx\) \(1.03334 - 0.464867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.588 + 1.28i)T \)
3 \( 1 + iT \)
23 \( 1 + (4.75 + 0.633i)T \)
good5 \( 1 - 2.28iT - 5T^{2} \)
7 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 - 0.215T + 13T^{2} \)
17 \( 1 - 0.222iT - 17T^{2} \)
19 \( 1 - 7.32T + 19T^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 + 3.33iT - 31T^{2} \)
37 \( 1 - 1.86iT - 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 - 2.67T + 43T^{2} \)
47 \( 1 + 6.03iT - 47T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + 3.93iT - 59T^{2} \)
61 \( 1 - 9.89iT - 61T^{2} \)
67 \( 1 + 1.94T + 67T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 4.15T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 + 16.3iT - 97T^{2} \)
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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

−11.51713858478543105130275532223, −11.10233116319567661173562962962, −10.03099853320347685736258135029, −8.958344478303990830673177627985, −7.80629885033664032381687331769, −7.27646310352248997606664846588, −5.67570756989824282572634038148, −4.16268447404553072570864781722, −2.80118166761162760382864464742, −1.50303478955859304120534783575, 1.34232750092510277097517508901, 4.10739789612718132919226511395, 5.03499250547633439722607078155, 5.74984100537116586064605730576, 7.39433187943319652722742263010, 8.202624497459455906537386601345, 9.063106693977373632940614495417, 9.744310421048105606709748275336, 10.99470399555304080336031053824, 11.84727395596920189767475201016

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