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Properties

Label 2-275-11.2-c2-0-21
Degree $2$
Conductor $275$
Sign $0.780 + 0.625i$
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  + (1.80 + 0.587i)2-s + (−3.61 + 2.62i)3-s + (−0.309 − 0.224i)4-s + (−8.09 + 2.62i)6-s + (4.89 − 6.74i)7-s + (−4.89 − 6.74i)8-s + (3.39 − 10.4i)9-s + (−10.3 − 3.66i)11-s + 1.70·12-s + (18.6 + 6.06i)13-s + (12.8 − 9.31i)14-s + (−4.42 − 13.6i)16-s + (20.4 − 6.65i)17-s + (12.2 − 16.9i)18-s + (−9.14 − 12.5i)19-s + ⋯
L(s)  = 1  + (0.904 + 0.293i)2-s + (−1.20 + 0.876i)3-s + (−0.0772 − 0.0561i)4-s + (−1.34 + 0.438i)6-s + (0.699 − 0.963i)7-s + (−0.612 − 0.842i)8-s + (0.377 − 1.16i)9-s + (−0.942 − 0.333i)11-s + 0.142·12-s + (1.43 + 0.466i)13-s + (0.916 − 0.665i)14-s + (−0.276 − 0.851i)16-s + (1.20 − 0.391i)17-s + (0.683 − 0.940i)18-s + (−0.481 − 0.662i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.780 + 0.625i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ 0.780 + 0.625i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29083 - 0.453618i\)
\(L(\frac12)\) \(\approx\) \(1.29083 - 0.453618i\)
\(L(2)\) 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 \( 1 + (-1.80 - 0.587i)T + (3.23 + 2.35i)T^{2} \)
3 \( 1 + (3.61 - 2.62i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (-4.89 + 6.74i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (-18.6 - 6.06i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-20.4 + 6.65i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (9.14 + 12.5i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 6.50T + 529T^{2} \)
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 \( 1 + (36.5 + 26.5i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (17.0 + 23.4i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 65.3iT - 1.84e3T^{2} \)
47 \( 1 + (-13.7 + 10.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (25.4 - 78.3i)T + (-2.27e3 - 1.65e3i)T^{2} \)
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 \( 1 - 76.0T + 4.48e3T^{2} \)
71 \( 1 + (33.6 + 103. i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (69.9 - 96.3i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-80.1 - 26.0i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (72.9 - 23.6i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 53.7T + 7.92e3T^{2} \)
97 \( 1 + (-50.4 + 155. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.42822189978234670352076406691, −10.73825989763936047042029947285, −10.07442946065220976199258216504, −8.778350966946626281775965825307, −7.32342055625197395712658957524, −6.07199119843734315927206625843, −5.35519628965900254631252387076, −4.49438602323308996145612493370, −3.65066777362257348243694133519, −0.65012964774728400070794961290, 1.58202439756417858782572512483, 3.23324196691708604863169496072, 4.93374736497932833476606878253, 5.56540264470854136631659859956, 6.32848952734508977942745227678, 7.938657632020006901287786707758, 8.540372642556213266637057525337, 10.35061012843762278182123278825, 11.28257097650198177905711630639, 11.94817609082221956292448797822

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