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Properties

Label 2-275-55.4-c1-0-4
Degree $2$
Conductor $275$
Sign $-0.228 - 0.973i$
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  + (−1.50 + 2.07i)2-s + (1.70 + 0.553i)3-s + (−1.41 − 4.34i)4-s + (−3.71 + 2.69i)6-s + (3.61 − 1.17i)7-s + (6.25 + 2.03i)8-s + (0.172 + 0.125i)9-s + (3.19 + 0.890i)11-s − 8.18i·12-s + (−2.64 + 3.63i)13-s + (−3.00 + 9.26i)14-s + (−6.24 + 4.53i)16-s + (2.94 + 4.05i)17-s + (−0.520 + 0.168i)18-s + (1.06 − 3.29i)19-s + ⋯
L(s)  = 1  + (−1.06 + 1.46i)2-s + (0.984 + 0.319i)3-s + (−0.705 − 2.17i)4-s + (−1.51 + 1.10i)6-s + (1.36 − 0.443i)7-s + (2.21 + 0.718i)8-s + (0.0575 + 0.0418i)9-s + (0.963 + 0.268i)11-s − 2.36i·12-s + (−0.733 + 1.00i)13-s + (−0.804 + 2.47i)14-s + (−1.56 + 1.13i)16-s + (0.714 + 0.983i)17-s + (−0.122 + 0.0398i)18-s + (0.245 − 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690912 + 0.872109i\)
\(L(\frac12)\) \(\approx\) \(0.690912 + 0.872109i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 - 2.07i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.70 - 0.553i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-3.61 + 1.17i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.64 - 3.63i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.94 - 4.05i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.06 + 3.29i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.105iT - 23T^{2} \)
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 \( 1 + (5.32 - 1.73i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.283 + 0.872i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.46iT - 43T^{2} \)
47 \( 1 + (1.19 + 0.387i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.23 - 4.44i)T + (-16.3 - 50.4i)T^{2} \)
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 \( 1 + 7.84iT - 67T^{2} \)
71 \( 1 + (-1.76 + 1.28i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.51 - 2.76i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.6 - 8.45i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.76 - 3.81i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.172T + 89T^{2} \)
97 \( 1 + (-2.62 + 3.60i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.98227681331336327605324182456, −10.83058927030793067203504037698, −9.697787706516698128319547009063, −9.056878879866865117141547326381, −8.300589560177729055390355055646, −7.52025915574672432383008508365, −6.64320649286146182538364055784, −5.22122485767569207363129850148, −4.08287254712999864756844026988, −1.66172868203612672412585004910, 1.40939439631007733408493097753, 2.52768999716686690903569483838, 3.55639606111170845414864348508, 5.20539406878363757963061716024, 7.52741896372073991047947564288, 8.077957830000254947291985144050, 8.841578559802135610886573580922, 9.635621172882501441278780177004, 10.66385361851648578171651839634, 11.65672205864489682027423438869

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