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Properties

Label 2-276-4.3-c2-0-19
Degree $2$
Conductor $276$
Sign $0.906 + 0.421i$
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  + (−0.431 + 1.95i)2-s + 1.73i·3-s + (−3.62 − 1.68i)4-s − 9.02·5-s + (−3.38 − 0.747i)6-s + 1.43i·7-s + (4.85 − 6.35i)8-s − 2.99·9-s + (3.89 − 17.6i)10-s + 4.88i·11-s + (2.91 − 6.28i)12-s + 2.77·13-s + (−2.79 − 0.618i)14-s − 15.6i·15-s + (10.3 + 12.2i)16-s + 20.6·17-s + ⋯
L(s)  = 1  + (−0.215 + 0.976i)2-s + 0.577i·3-s + (−0.906 − 0.421i)4-s − 1.80·5-s + (−0.563 − 0.124i)6-s + 0.204i·7-s + (0.607 − 0.794i)8-s − 0.333·9-s + (0.389 − 1.76i)10-s + 0.443i·11-s + (0.243 − 0.523i)12-s + 0.213·13-s + (−0.199 − 0.0441i)14-s − 1.04i·15-s + (0.644 + 0.764i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.906 + 0.421i$
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: \((2,\ 276,\ (\ :1),\ 0.906 + 0.421i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.417470 - 0.0922468i\)
\(L(\frac12)\) \(\approx\) \(0.417470 - 0.0922468i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.431 - 1.95i)T \)
3 \( 1 - 1.73iT \)
23 \( 1 + 4.79iT \)
good5 \( 1 + 9.02T + 25T^{2} \)
7 \( 1 - 1.43iT - 49T^{2} \)
11 \( 1 - 4.88iT - 121T^{2} \)
13 \( 1 - 2.77T + 169T^{2} \)
17 \( 1 - 20.6T + 289T^{2} \)
19 \( 1 + 34.2iT - 361T^{2} \)
29 \( 1 + 44.4T + 841T^{2} \)
31 \( 1 + 27.2iT - 961T^{2} \)
37 \( 1 + 8.67T + 1.36e3T^{2} \)
41 \( 1 + 71.6T + 1.68e3T^{2} \)
43 \( 1 + 64.5iT - 1.84e3T^{2} \)
47 \( 1 - 58.2iT - 2.20e3T^{2} \)
53 \( 1 - 32.8T + 2.80e3T^{2} \)
59 \( 1 + 69.4iT - 3.48e3T^{2} \)
61 \( 1 + 10.0T + 3.72e3T^{2} \)
67 \( 1 - 2.46iT - 4.48e3T^{2} \)
71 \( 1 + 59.3iT - 5.04e3T^{2} \)
73 \( 1 + 33.9T + 5.32e3T^{2} \)
79 \( 1 + 1.10iT - 6.24e3T^{2} \)
83 \( 1 + 58.2iT - 6.88e3T^{2} \)
89 \( 1 - 14.4T + 7.92e3T^{2} \)
97 \( 1 + 46.0T + 9.40e3T^{2} \)
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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

−11.53492104453870693610075647151, −10.62116221253408350779627164295, −9.409284948077578036556250494980, −8.574761705015230279818403223079, −7.66509106941775123804200989834, −6.95454447158892358625442258974, −5.39529164630695257570206598727, −4.42611626165697127950731553159, −3.47628174592396521434375141622, −0.27214369320208122493443969622, 1.25769646490534847223335879821, 3.31684702613655768618898478098, 3.90417014598741529675861825244, 5.45787541099043004571692596573, 7.27028720070709064825293856850, 8.003357004112923106002790131418, 8.624976591249833487316690787888, 10.08878808129930067849618092601, 11.00283553960314941664824570466, 11.91718366227636114577240139921

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