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Properties

Label 2-275-11.3-c1-0-15
Degree $2$
Conductor $275$
Sign $-0.639 + 0.768i$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.07 − 1.50i)2-s + (−0.553 − 1.70i)3-s + (1.41 − 4.34i)4-s + (−3.71 − 2.69i)6-s + (−1.17 + 3.61i)7-s + (−2.03 − 6.25i)8-s + (−0.172 + 0.125i)9-s + (3.19 − 0.890i)11-s − 8.18·12-s + (−3.63 + 2.64i)13-s + (3.00 + 9.26i)14-s + (−6.24 − 4.53i)16-s + (4.05 + 2.94i)17-s + (−0.168 + 0.520i)18-s + (−1.06 − 3.29i)19-s + ⋯
L(s)  = 1  + (1.46 − 1.06i)2-s + (−0.319 − 0.984i)3-s + (0.705 − 2.17i)4-s + (−1.51 − 1.10i)6-s + (−0.443 + 1.36i)7-s + (−0.718 − 2.21i)8-s + (−0.0575 + 0.0418i)9-s + (0.963 − 0.268i)11-s − 2.36·12-s + (−1.00 + 0.733i)13-s + (0.804 + 2.47i)14-s + (−1.56 − 1.13i)16-s + (0.983 + 0.714i)17-s + (−0.0398 + 0.122i)18-s + (−0.245 − 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.639 + 0.768i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.639 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02316 - 2.18374i\)
\(L(\frac12)\) \(\approx\) \(1.02316 - 2.18374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.19 + 0.890i)T \)
good2 \( 1 + (-2.07 + 1.50i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.553 + 1.70i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.17 - 3.61i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.63 - 2.64i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.05 - 2.94i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.06 + 3.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.105T + 23T^{2} \)
29 \( 1 + (0.726 - 2.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.83 - 2.78i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.73 + 5.32i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.283 - 0.872i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.46T + 43T^{2} \)
47 \( 1 + (0.387 + 1.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.44 - 3.23i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.365 - 1.12i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.92 - 4.30i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.84T + 67T^{2} \)
71 \( 1 + (-1.76 - 1.28i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.76 - 8.51i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.6 - 8.45i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.81 + 2.76i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.172T + 89T^{2} \)
97 \( 1 + (3.60 - 2.62i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.91030320652992412921386197851, −11.25146301463371509131656493372, −9.882251000364581549532973807141, −8.967084663158530574364015087407, −7.11286585342101886910209843504, −6.18093127398228445275455792009, −5.46736988372488491759026606899, −4.07931480986337842747104042348, −2.72236115866014692684227275618, −1.60330772528169182872829917777, 3.37327223467458579557695856208, 4.16580781866865530510225007927, 4.96587163865869574921328073392, 6.03234700612722333432957593989, 7.19214940934373837519309007411, 7.74771817790273586945605123098, 9.612853798194663454484106824060, 10.30211101509067373108616118201, 11.55045504772262174748348712795, 12.49535454324077357361293576736

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