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Properties

Label 2-275-5.2-c2-0-20
Degree $2$
Conductor $275$
Sign $0.793 - 0.608i$
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  + (1.94 + 1.94i)2-s + (2.32 − 2.32i)3-s + 3.57i·4-s + 9.06·6-s + (4.69 + 4.69i)7-s + (0.833 − 0.833i)8-s − 1.85i·9-s − 3.31·11-s + (8.32 + 8.32i)12-s + (8.25 − 8.25i)13-s + 18.2i·14-s + 17.5·16-s + (−8.51 − 8.51i)17-s + (3.61 − 3.61i)18-s − 1.38i·19-s + ⋯
L(s)  = 1  + (0.972 + 0.972i)2-s + (0.776 − 0.776i)3-s + 0.892i·4-s + 1.51·6-s + (0.671 + 0.671i)7-s + (0.104 − 0.104i)8-s − 0.206i·9-s − 0.301·11-s + (0.693 + 0.693i)12-s + (0.634 − 0.634i)13-s + 1.30i·14-s + 1.09·16-s + (−0.500 − 0.500i)17-s + (0.200 − 0.200i)18-s − 0.0730i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.793 - 0.608i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ 0.793 - 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.38351 + 1.14808i\)
\(L(\frac12)\) \(\approx\) \(3.38351 + 1.14808i\)
\(L(2)\) 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 \( 1 + 3.31T \)
good2 \( 1 + (-1.94 - 1.94i)T + 4iT^{2} \)
3 \( 1 + (-2.32 + 2.32i)T - 9iT^{2} \)
7 \( 1 + (-4.69 - 4.69i)T + 49iT^{2} \)
13 \( 1 + (-8.25 + 8.25i)T - 169iT^{2} \)
17 \( 1 + (8.51 + 8.51i)T + 289iT^{2} \)
19 \( 1 + 1.38iT - 361T^{2} \)
23 \( 1 + (29.4 - 29.4i)T - 529iT^{2} \)
29 \( 1 + 18.7iT - 841T^{2} \)
31 \( 1 - 11.9T + 961T^{2} \)
37 \( 1 + (13.2 + 13.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 3.05T + 1.68e3T^{2} \)
43 \( 1 + (37.6 - 37.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (59.2 + 59.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (59.2 - 59.2i)T - 2.80e3iT^{2} \)
59 \( 1 + 89.9iT - 3.48e3T^{2} \)
61 \( 1 + 93.2T + 3.72e3T^{2} \)
67 \( 1 + (34.1 + 34.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 112.T + 5.04e3T^{2} \)
73 \( 1 + (-45.0 + 45.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 120. iT - 6.24e3T^{2} \)
83 \( 1 + (87.1 - 87.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 47.4iT - 7.92e3T^{2} \)
97 \( 1 + (-11.1 - 11.1i)T + 9.40e3iT^{2} \)
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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

−12.16556840394493043167925745198, −11.06107377961485902413290684640, −9.661373546363680543135716526354, −8.196113421436646949831102308493, −7.946488903581185795921605802695, −6.77585814438601422764213853019, −5.73144760923178361663320495335, −4.81851510241130289899678065411, −3.32576606578302550995886935611, −1.83402569107202908278241418346, 1.78409015920732492432661842331, 3.17016265436593523439993371445, 4.15643809190249533147934264213, 4.71581458217171288891714989913, 6.34055039348797252958069360276, 7.970541783352008781135359148682, 8.748738825506857235734465499194, 10.08727349666778742462041619194, 10.67011500431583954025046850413, 11.55825559455126860797113585187

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