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Properties

Label 2-276-23.13-c1-0-2
Degree $2$
Conductor $276$
Sign $0.936 + 0.351i$
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  + (0.841 + 0.540i)3-s + (1.11 − 2.44i)5-s + (0.161 + 0.0474i)7-s + (0.415 + 0.909i)9-s + (3.48 − 4.02i)11-s + (−3.68 + 1.08i)13-s + (2.26 − 1.45i)15-s + (0.947 + 6.58i)17-s + (0.980 − 6.82i)19-s + (0.110 + 0.127i)21-s + (2.24 + 4.23i)23-s + (−1.45 − 1.68i)25-s + (−0.142 + 0.989i)27-s + (0.0811 + 0.564i)29-s + (−5.85 + 3.76i)31-s + ⋯
L(s)  = 1  + (0.485 + 0.312i)3-s + (0.499 − 1.09i)5-s + (0.0610 + 0.0179i)7-s + (0.138 + 0.303i)9-s + (1.05 − 1.21i)11-s + (−1.02 + 0.300i)13-s + (0.583 − 0.375i)15-s + (0.229 + 1.59i)17-s + (0.225 − 1.56i)19-s + (0.0240 + 0.0277i)21-s + (0.467 + 0.884i)23-s + (−0.291 − 0.336i)25-s + (−0.0273 + 0.190i)27-s + (0.0150 + 0.104i)29-s + (−1.05 + 0.675i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.936 + 0.351i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.936 + 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58538 - 0.287414i\)
\(L(\frac12)\) \(\approx\) \(1.58538 - 0.287414i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (-2.24 - 4.23i)T \)
good5 \( 1 + (-1.11 + 2.44i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (-0.161 - 0.0474i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-3.48 + 4.02i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (3.68 - 1.08i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.947 - 6.58i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.980 + 6.82i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-0.0811 - 0.564i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (5.85 - 3.76i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-3.67 - 8.05i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-2.02 + 4.42i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (5.68 + 3.65i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 2.26T + 47T^{2} \)
53 \( 1 + (4.67 + 1.37i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (2.18 - 0.642i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (11.7 - 7.54i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-1.05 - 1.21i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-6.21 - 7.17i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.545 - 3.79i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (2.51 - 0.739i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (3.72 + 8.15i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-5.28 - 3.39i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (5.51 - 12.0i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.85109498077557996573540455389, −10.89377188232820338473904063116, −9.623797966287856752018439477775, −9.005650413564082554839618213707, −8.316905106499003551110870854148, −6.89967561043191250388869178313, −5.61348541821468747867173644125, −4.62809831427140684221929245597, −3.32226860168900896409408643761, −1.50593557442672616171905596312, 2.00951833948540967500119361299, 3.15459172001752069989502778727, 4.66900370327337662724384794443, 6.19275771757095714997434600306, 7.13110563344494796384776349655, 7.76383228666568504599177009538, 9.509272634781130977653371120759, 9.715396759125896475715314254190, 10.96297771540457727019002296954, 12.07640433973003838246135402002

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