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Properties

Label 2-275-55.27-c2-0-29
Degree $2$
Conductor $275$
Sign $-0.397 + 0.917i$
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  + (1.35 − 2.66i)2-s + (5.64 − 0.893i)3-s + (−2.89 − 3.98i)4-s + (5.27 − 16.2i)6-s + (−7.26 − 1.14i)7-s + (−2.74 + 0.434i)8-s + (22.4 − 7.30i)9-s + (5.78 + 9.35i)11-s + (−19.9 − 19.9i)12-s + (−0.739 − 0.376i)13-s + (−12.9 + 17.7i)14-s + (3.52 − 10.8i)16-s + (−23.0 + 11.7i)17-s + (11.0 − 69.7i)18-s + (−6.24 + 8.59i)19-s + ⋯
L(s)  = 1  + (0.678 − 1.33i)2-s + (1.88 − 0.297i)3-s + (−0.724 − 0.997i)4-s + (0.879 − 2.70i)6-s + (−1.03 − 0.164i)7-s + (−0.343 + 0.0543i)8-s + (2.49 − 0.811i)9-s + (0.525 + 0.850i)11-s + (−1.65 − 1.65i)12-s + (−0.0568 − 0.0289i)13-s + (−0.922 + 1.26i)14-s + (0.220 − 0.678i)16-s + (−1.35 + 0.690i)17-s + (0.613 − 3.87i)18-s + (−0.328 + 0.452i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.397 + 0.917i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.08543 - 3.17540i\)
\(L(\frac12)\) \(\approx\) \(2.08543 - 3.17540i\)
\(L(2)\) 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 \( 1 + (-5.78 - 9.35i)T \)
good2 \( 1 + (-1.35 + 2.66i)T + (-2.35 - 3.23i)T^{2} \)
3 \( 1 + (-5.64 + 0.893i)T + (8.55 - 2.78i)T^{2} \)
7 \( 1 + (7.26 + 1.14i)T + (46.6 + 15.1i)T^{2} \)
13 \( 1 + (0.739 + 0.376i)T + (99.3 + 136. i)T^{2} \)
17 \( 1 + (23.0 - 11.7i)T + (169. - 233. i)T^{2} \)
19 \( 1 + (6.24 - 8.59i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + (15.1 - 15.1i)T - 529iT^{2} \)
29 \( 1 + (1.59 + 2.19i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (2.08 + 6.42i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-29.9 - 4.75i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (-47.7 - 34.7i)T + (519. + 1.59e3i)T^{2} \)
43 \( 1 + (19.6 - 19.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (14.1 + 89.6i)T + (-2.10e3 + 682. i)T^{2} \)
53 \( 1 + (-3.94 - 2.01i)T + (1.65e3 + 2.27e3i)T^{2} \)
59 \( 1 + (-34.3 - 47.2i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-6.01 + 18.5i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + (-15.8 - 15.8i)T + 4.48e3iT^{2} \)
71 \( 1 + (19.5 - 60.1i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (2.76 - 17.4i)T + (-5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (-21.7 + 7.06i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (38.0 + 74.6i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + 64.7iT - 7.92e3T^{2} \)
97 \( 1 + (17.7 - 34.8i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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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

−11.60289767181605420699061156739, −10.13704576244287461689191567010, −9.727107712900628547360471383396, −8.774762342088478440374268759368, −7.57916923321259953011194882338, −6.53633369710799014191186356823, −4.28393137023834391163691069962, −3.72625078418922006504684411094, −2.60107580167702108520022941047, −1.70770358848100096985300727947, 2.53678570473645899543721458836, 3.71291844966727652812749631989, 4.58657326084621771640703580459, 6.25591552944326578775831493491, 7.01748160105582637301765498169, 8.062062693010786942664797346870, 8.897111356202327095106496623531, 9.496771015719815829907319620773, 10.82823791822751976151856359167, 12.68411825129887617897091757855

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