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Properties

Label 2-275-11.9-c1-0-0
Degree $2$
Conductor $275$
Sign $-0.119 + 0.992i$
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  + (−0.511 + 1.57i)2-s + (−1.59 + 1.16i)3-s + (−0.596 − 0.433i)4-s + (−1.00 − 3.10i)6-s + (−1.81 − 1.31i)7-s + (−1.68 + 1.22i)8-s + (0.278 − 0.855i)9-s + (−3.27 − 0.547i)11-s + 1.45·12-s + (1.14 − 3.51i)13-s + (3.00 − 2.18i)14-s + (−1.52 − 4.68i)16-s + (0.687 + 2.11i)17-s + (1.20 + 0.875i)18-s + (−4.27 + 3.10i)19-s + ⋯
L(s)  = 1  + (−0.361 + 1.11i)2-s + (−0.922 + 0.670i)3-s + (−0.298 − 0.216i)4-s + (−0.412 − 1.26i)6-s + (−0.685 − 0.498i)7-s + (−0.597 + 0.434i)8-s + (0.0926 − 0.285i)9-s + (−0.986 − 0.165i)11-s + 0.420·12-s + (0.317 − 0.975i)13-s + (0.802 − 0.583i)14-s + (−0.380 − 1.17i)16-s + (0.166 + 0.513i)17-s + (0.283 + 0.206i)18-s + (−0.981 + 0.712i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.119 + 0.992i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.119 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123028 - 0.138757i\)
\(L(\frac12)\) \(\approx\) \(0.123028 - 0.138757i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.27 + 0.547i)T \)
good2 \( 1 + (0.511 - 1.57i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.59 - 1.16i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (1.81 + 1.31i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.14 + 3.51i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.687 - 2.11i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.27 - 3.10i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.85T + 23T^{2} \)
29 \( 1 + (0.152 + 0.110i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.212 + 0.653i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.09 - 1.52i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.40 - 4.65i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.41T + 43T^{2} \)
47 \( 1 + (9.71 - 7.06i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.91 - 12.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.278 - 0.202i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.535 - 1.64i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.650T + 67T^{2} \)
71 \( 1 + (-1.43 - 4.42i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.16 + 5.20i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.23 - 6.88i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.983 + 3.02i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 9.92T + 89T^{2} \)
97 \( 1 + (0.700 - 2.15i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.64507733728898579367134937155, −11.38770451005304045482565127643, −10.52211994539608403491122279013, −9.909281544994881159023107507125, −8.459960417605195503821501532150, −7.74946536301080325961229921615, −6.47348045755523369165872708481, −5.80426608386445770571168458129, −4.81566606639772325824069793632, −3.17093347858415034607574511870, 0.16523906543227639295969956767, 1.95052029262294170435337978876, 3.25025101284325968383040189065, 5.07824769365075705995123719825, 6.35300226811262709391551763232, 6.92680169970833622133850224555, 8.631832133764711608632374519930, 9.522171315384965414406919078948, 10.49061467833532125405793763558, 11.38762469048047728415715497663

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