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Properties

Label 2-275-5.3-c2-0-8
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  + (−0.925 + 0.925i)2-s + (−3.86 − 3.86i)3-s + 2.28i·4-s + 7.15·6-s + (−2.48 + 2.48i)7-s + (−5.81 − 5.81i)8-s + 20.9i·9-s − 3.31·11-s + (8.85 − 8.85i)12-s + (11.1 + 11.1i)13-s − 4.60i·14-s + 1.60·16-s + (−1.87 + 1.87i)17-s + (−19.3 − 19.3i)18-s − 32.2i·19-s + ⋯
L(s)  = 1  + (−0.462 + 0.462i)2-s + (−1.28 − 1.28i)3-s + 0.572i·4-s + 1.19·6-s + (−0.355 + 0.355i)7-s + (−0.727 − 0.727i)8-s + 2.32i·9-s − 0.301·11-s + (0.737 − 0.737i)12-s + (0.857 + 0.857i)13-s − 0.328i·14-s + 0.100·16-s + (−0.110 + 0.110i)17-s + (−1.07 − 1.07i)18-s − 1.69i·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: $\chi_{275} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.561861 - 0.190649i\)
\(L(\frac12)\) \(\approx\) \(0.561861 - 0.190649i\)
\(L(2)\) 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 \( 1 + (0.925 - 0.925i)T - 4iT^{2} \)
3 \( 1 + (3.86 + 3.86i)T + 9iT^{2} \)
7 \( 1 + (2.48 - 2.48i)T - 49iT^{2} \)
13 \( 1 + (-11.1 - 11.1i)T + 169iT^{2} \)
17 \( 1 + (1.87 - 1.87i)T - 289iT^{2} \)
19 \( 1 + 32.2iT - 361T^{2} \)
23 \( 1 + (10.9 + 10.9i)T + 529iT^{2} \)
29 \( 1 + 15.8iT - 841T^{2} \)
31 \( 1 - 56.3T + 961T^{2} \)
37 \( 1 + (-43.0 + 43.0i)T - 1.36e3iT^{2} \)
41 \( 1 - 41.8T + 1.68e3T^{2} \)
43 \( 1 + (3.70 + 3.70i)T + 1.84e3iT^{2} \)
47 \( 1 + (27.8 - 27.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (14.7 + 14.7i)T + 2.80e3iT^{2} \)
59 \( 1 + 102. iT - 3.48e3T^{2} \)
61 \( 1 + 34.3T + 3.72e3T^{2} \)
67 \( 1 + (6.48 - 6.48i)T - 4.48e3iT^{2} \)
71 \( 1 + 20.3T + 5.04e3T^{2} \)
73 \( 1 + (-44.6 - 44.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 28.9iT - 6.24e3T^{2} \)
83 \( 1 + (-99.4 - 99.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 143. iT - 7.92e3T^{2} \)
97 \( 1 + (4.67 - 4.67i)T - 9.40e3iT^{2} \)
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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

−11.58654966124189484171120427147, −11.03953410423316233306970801892, −9.484944033308909450411540958964, −8.382935845889439076051467388105, −7.50818693115572215700547065136, −6.51077183366991873802114809487, −6.14321200537948279431815616637, −4.55056304868303582197836257819, −2.49935720040292970660001456376, −0.57164020459559463119127742499, 0.927100365298370386452514282240, 3.36221686409048086699765393861, 4.63506192676191603686256914227, 5.77375744590525114527463553313, 6.25020674357795894221303681270, 8.184370006253825603425011229023, 9.446664991054220890929372538736, 10.26426510313626516449467949671, 10.46901559804515575180576151369, 11.48209444919775414496865744726

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