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Properties

Label 2-275-55.4-c1-0-6
Degree $2$
Conductor $275$
Sign $0.120 - 0.992i$
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  + (−0.433 + 0.596i)2-s + (2.67 + 0.868i)3-s + (0.449 + 1.38i)4-s + (−1.67 + 1.21i)6-s + (−0.980 + 0.318i)7-s + (−2.42 − 0.787i)8-s + (3.96 + 2.88i)9-s + (1.93 − 2.69i)11-s + 4.09i·12-s + (−2.02 + 2.79i)13-s + (0.235 − 0.723i)14-s + (−0.834 + 0.606i)16-s + (−1.40 − 1.94i)17-s + (−3.44 + 1.11i)18-s + (2.36 − 7.29i)19-s + ⋯
L(s)  = 1  + (−0.306 + 0.421i)2-s + (1.54 + 0.501i)3-s + (0.224 + 0.692i)4-s + (−0.684 + 0.497i)6-s + (−0.370 + 0.120i)7-s + (−0.857 − 0.278i)8-s + (1.32 + 0.961i)9-s + (0.583 − 0.811i)11-s + 1.18i·12-s + (−0.562 + 0.773i)13-s + (0.0628 − 0.193i)14-s + (−0.208 + 0.151i)16-s + (−0.341 − 0.470i)17-s + (−0.811 + 0.263i)18-s + (0.543 − 1.67i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.120 - 0.992i$
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: \((2,\ 275,\ (\ :1/2),\ 0.120 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29670 + 1.14859i\)
\(L(\frac12)\) \(\approx\) \(1.29670 + 1.14859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 2.69i)T \)
good2 \( 1 + (0.433 - 0.596i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.67 - 0.868i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.980 - 0.318i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.02 - 2.79i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.40 + 1.94i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.36 + 7.29i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.45iT - 23T^{2} \)
29 \( 1 + (-1.83 - 5.66i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.98 - 2.16i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (5.66 - 1.84i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.21 + 3.74i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.64iT - 43T^{2} \)
47 \( 1 + (-5.55 - 1.80i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.96 + 9.58i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.910 + 2.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.00 - 1.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.14iT - 67T^{2} \)
71 \( 1 + (1.63 - 1.18i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.785 - 0.255i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.77 + 7.09i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.946 - 1.30i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.16T + 89T^{2} \)
97 \( 1 + (-1.43 + 1.97i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.15571083792512274130018538454, −11.15141656003221163133711948241, −9.701539502991045562466168892760, −8.949922136430979784858121805578, −8.586853255311971377493549588860, −7.33585267068394219910925364390, −6.61976737855160213165409667892, −4.63254426480577726181498267889, −3.37174242496646949265376430468, −2.61644220540105824637604222727, 1.56910256551485570150278275458, 2.65994058571815533326523097717, 3.91490782899396041680177567326, 5.75772080995362184082585502275, 6.99788016541028212328277950716, 7.909689054319357506798823551724, 8.917462203067238784261535733052, 9.834933803408437989694293011489, 10.25937904441000380666537824297, 11.86249545037669069862247655377

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