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Properties

Label 2-276-92.91-c1-0-3
Degree $2$
Conductor $276$
Sign $-0.616 - 0.787i$
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  + (0.714 + 1.22i)2-s − i·3-s + (−0.979 + 1.74i)4-s + 3.78i·5-s + (1.22 − 0.714i)6-s − 1.02·7-s + (−2.82 + 0.0497i)8-s − 9-s + (−4.61 + 2.70i)10-s − 4.16·11-s + (1.74 + 0.979i)12-s + 5.75·13-s + (−0.730 − 1.24i)14-s + 3.78·15-s + (−2.08 − 3.41i)16-s + 2.80i·17-s + ⋯
L(s)  = 1  + (0.505 + 0.863i)2-s − 0.577i·3-s + (−0.489 + 0.871i)4-s + 1.69i·5-s + (0.498 − 0.291i)6-s − 0.386·7-s + (−0.999 + 0.0175i)8-s − 0.333·9-s + (−1.45 + 0.854i)10-s − 1.25·11-s + (0.503 + 0.282i)12-s + 1.59·13-s + (−0.195 − 0.333i)14-s + 0.976·15-s + (−0.520 − 0.854i)16-s + 0.680i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.616 - 0.787i$
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: \((2,\ 276,\ (\ :1/2),\ -0.616 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599982 + 1.23199i\)
\(L(\frac12)\) \(\approx\) \(0.599982 + 1.23199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.714 - 1.22i)T \)
3 \( 1 + iT \)
23 \( 1 + (-4.42 - 1.84i)T \)
good5 \( 1 - 3.78iT - 5T^{2} \)
7 \( 1 + 1.02T + 7T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
13 \( 1 - 5.75T + 13T^{2} \)
17 \( 1 - 2.80iT - 17T^{2} \)
19 \( 1 - 4.00T + 19T^{2} \)
29 \( 1 - 0.341T + 29T^{2} \)
31 \( 1 - 5.39iT - 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 - 8.31T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 + 2.69iT - 47T^{2} \)
53 \( 1 - 0.814iT - 53T^{2} \)
59 \( 1 - 2.67iT - 59T^{2} \)
61 \( 1 + 7.77iT - 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 - 2.36iT - 71T^{2} \)
73 \( 1 - 5.19T + 73T^{2} \)
79 \( 1 - 6.91T + 79T^{2} \)
83 \( 1 - 1.12T + 83T^{2} \)
89 \( 1 + 9.76iT - 89T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−12.53801595599644371090484064300, −11.18251920786671894011331057880, −10.62012894250074587784492422832, −9.138044124008770355165976136025, −7.888974511789882983719962421873, −7.23866062211789671884599230788, −6.31284912387652827183157990573, −5.58953241676853096577134126612, −3.64307100700030991543146278113, −2.79794802170063529910113842782, 0.958515423786601873151282890520, 2.97211238273991271544931574335, 4.30190738019820948781816660703, 5.13675390263237172379586536682, 5.95578343180594925559044764734, 8.061386310510962826373911096388, 9.023226797678095760330162025389, 9.621217734602518694458946782430, 10.73299831734814663523677164751, 11.56384778732487181131684583253

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