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Properties

Label 2-276-23.3-c1-0-0
Degree $2$
Conductor $276$
Sign $-0.612 - 0.790i$
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  + (0.415 − 0.909i)3-s + (−2.38 + 2.74i)5-s + (−3.67 + 2.36i)7-s + (−0.654 − 0.755i)9-s + (−0.389 + 2.70i)11-s + (−5.17 − 3.32i)13-s + (1.51 + 3.30i)15-s + (1.26 + 0.370i)17-s + (5.16 − 1.51i)19-s + (0.621 + 4.32i)21-s + (0.596 + 4.75i)23-s + (−1.17 − 8.14i)25-s + (−0.959 + 0.281i)27-s + (6.27 + 1.84i)29-s + (1.69 + 3.72i)31-s + ⋯
L(s)  = 1  + (0.239 − 0.525i)3-s + (−1.06 + 1.22i)5-s + (−1.38 + 0.892i)7-s + (−0.218 − 0.251i)9-s + (−0.117 + 0.816i)11-s + (−1.43 − 0.922i)13-s + (0.390 + 0.854i)15-s + (0.306 + 0.0898i)17-s + (1.18 − 0.348i)19-s + (0.135 + 0.943i)21-s + (0.124 + 0.992i)23-s + (−0.234 − 1.62i)25-s + (−0.184 + 0.0542i)27-s + (1.16 + 0.341i)29-s + (0.305 + 0.668i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.612 - 0.790i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245888 + 0.501532i\)
\(L(\frac12)\) \(\approx\) \(0.245888 + 0.501532i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-0.596 - 4.75i)T \)
good5 \( 1 + (2.38 - 2.74i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (3.67 - 2.36i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.389 - 2.70i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (5.17 + 3.32i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-1.26 - 0.370i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-5.16 + 1.51i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-6.27 - 1.84i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-1.69 - 3.72i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (3.77 + 4.35i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (5.09 - 5.87i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.90 - 8.56i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 5.07T + 47T^{2} \)
53 \( 1 + (-3.43 + 2.20i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (2.90 + 1.86i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (1.10 + 2.41i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.627 - 4.36i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-1.45 - 10.1i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-6.23 + 1.83i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (2.84 + 1.82i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-4.71 - 5.43i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (4.07 - 8.92i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (1.60 - 1.85i)T + (-13.8 - 96.0i)T^{2} \)
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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

−12.24104433659019740000359327753, −11.55095398531977737842359616490, −10.14065430735022172786602307321, −9.593561787914021014076805205285, −8.099539256954351779733111723343, −7.23415777614389395090377617417, −6.64616046766234063971734312196, −5.20635976544185872517271446636, −3.25921885468707002661466687069, −2.79340846305641049024683917243, 0.39755783949839541115845281630, 3.19380228010814800012293250657, 4.17104273612351105428864926023, 5.11533775428190005116085708585, 6.73500242387827746214895718863, 7.75377778148302965027622010403, 8.753214388146686749469624149480, 9.643736564403741992227520045392, 10.38254853943243999186973263981, 11.86515947962633691700381254996

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