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Properties

Label 2-276-23.8-c1-0-3
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: $\chi_{276} (169, \cdot )$
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

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

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