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Properties

Label 2-275-55.24-c2-0-10
Degree $2$
Conductor $275$
Sign $-0.977 - 0.210i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
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  + (−0.587 + 1.80i)2-s + (2.62 + 3.61i)3-s + (0.309 + 0.224i)4-s + (−8.09 + 2.62i)6-s + (6.74 + 4.89i)7-s + (−6.74 + 4.89i)8-s + (−3.39 + 10.4i)9-s + (−10.3 − 3.66i)11-s + 1.70i·12-s + (6.06 − 18.6i)13-s + (−12.8 + 9.31i)14-s + (−4.42 − 13.6i)16-s + (6.65 + 20.4i)17-s + (−16.9 − 12.2i)18-s + (9.14 + 12.5i)19-s + ⋯
L(s)  = 1  + (−0.293 + 0.904i)2-s + (0.876 + 1.20i)3-s + (0.0772 + 0.0561i)4-s + (−1.34 + 0.438i)6-s + (0.963 + 0.699i)7-s + (−0.842 + 0.612i)8-s + (−0.377 + 1.16i)9-s + (−0.942 − 0.333i)11-s + 0.142i·12-s + (0.466 − 1.43i)13-s + (−0.916 + 0.665i)14-s + (−0.276 − 0.851i)16-s + (0.391 + 1.20i)17-s + (−0.940 − 0.683i)18-s + (0.481 + 0.662i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.977 - 0.210i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.215219 + 2.02066i\)
\(L(\frac12)\) \(\approx\) \(0.215219 + 2.02066i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (10.3 + 3.66i)T \)
good2 \( 1 + (0.587 - 1.80i)T + (-3.23 - 2.35i)T^{2} \)
3 \( 1 + (-2.62 - 3.61i)T + (-2.78 + 8.55i)T^{2} \)
7 \( 1 + (-6.74 - 4.89i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-6.06 + 18.6i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-6.65 - 20.4i)T + (-233. + 169. i)T^{2} \)
19 \( 1 + (-9.14 - 12.5i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 6.50iT - 529T^{2} \)
29 \( 1 + (2.70 - 3.71i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-9.93 + 30.5i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-26.5 + 36.5i)T + (-423. - 1.30e3i)T^{2} \)
41 \( 1 + (17.0 + 23.4i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 65.3T + 1.84e3T^{2} \)
47 \( 1 + (-10.0 - 13.7i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-78.3 - 25.4i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-1.25 - 0.910i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (24.0 - 7.80i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 76.0iT - 4.48e3T^{2} \)
71 \( 1 + (33.6 + 103. i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-96.3 - 69.9i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (80.1 + 26.0i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-23.6 - 72.9i)T + (-5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 53.7T + 7.92e3T^{2} \)
97 \( 1 + (-155. - 50.4i)T + (7.61e3 + 5.53e3i)T^{2} \)
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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

−12.01840650299319838945606341979, −10.84573869697307278286465315648, −10.13959176898028154647147311473, −8.867410411499346035770202004927, −8.176628706178119292069111652900, −7.81995167611410569016417137367, −5.87906841360844683075843224622, −5.24102298843490039760469510238, −3.64465320608412521530805678649, −2.53460410969863774482489741283, 1.09776519577493656314500060971, 2.04790638932904569083451720012, 3.15138571170851906729696295074, 4.85214272783208199145995876953, 6.69071493920721327435539605928, 7.34982544426682092506853589401, 8.337236196326583496106659408408, 9.335324832160185551079679480120, 10.35161842608706467794897572115, 11.50428270508777244488643272097

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