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Properties

Label 2-275-55.14-c1-0-7
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: $\chi_{275} (124, \cdot )$
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

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

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