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Properties

Label 2-275-55.9-c1-0-12
Degree $2$
Conductor $275$
Sign $0.893 + 0.448i$
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  + (1.99 + 0.647i)2-s + (−1.12 − 1.54i)3-s + (1.93 + 1.40i)4-s + (−1.23 − 3.81i)6-s + (1.80 − 2.48i)7-s + (0.480 + 0.661i)8-s + (−0.203 + 0.627i)9-s + (1.86 + 2.74i)11-s − 4.57i·12-s + (2.90 + 0.942i)13-s + (5.19 − 3.77i)14-s + (−0.947 − 2.91i)16-s + (−0.441 + 0.143i)17-s + (−0.812 + 1.11i)18-s + (−6.38 + 4.64i)19-s + ⋯
L(s)  = 1  + (1.40 + 0.457i)2-s + (−0.649 − 0.893i)3-s + (0.966 + 0.702i)4-s + (−0.505 − 1.55i)6-s + (0.681 − 0.937i)7-s + (0.169 + 0.233i)8-s + (−0.0679 + 0.209i)9-s + (0.561 + 0.827i)11-s − 1.31i·12-s + (0.804 + 0.261i)13-s + (1.38 − 1.00i)14-s + (−0.236 − 0.729i)16-s + (−0.106 + 0.0347i)17-s + (−0.191 + 0.263i)18-s + (−1.46 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.893 + 0.448i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17293 - 0.514276i\)
\(L(\frac12)\) \(\approx\) \(2.17293 - 0.514276i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 2.74i)T \)
good2 \( 1 + (-1.99 - 0.647i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.12 + 1.54i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.80 + 2.48i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-2.90 - 0.942i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.441 - 0.143i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.38 - 4.64i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.39iT - 23T^{2} \)
29 \( 1 + (-3.01 - 2.18i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.23 - 9.96i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.08 - 1.49i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.56 + 2.59i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.31iT - 43T^{2} \)
47 \( 1 + (1.75 + 2.41i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.98 - 1.29i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.27 - 1.65i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.623 + 1.91i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.75iT - 67T^{2} \)
71 \( 1 + (2.01 + 6.20i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.80 - 7.98i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.57 + 10.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (8.48 - 2.75i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (14.6 + 4.74i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.28896346906693301395195506397, −11.27091653131455433272033012366, −10.29993871690621302212545759885, −8.641716287173403724506835591630, −7.27325129646821388427991231179, −6.72693264031871795371429058790, −5.87547541515525959337852122070, −4.59840873532573754761558205489, −3.79686080113750240767114707040, −1.56066121785546956610081794654, 2.34710766505392959803749112037, 3.85017832215660411204497940471, 4.66610235084630373713312564905, 5.64092446088590283413933052869, 6.25705685901129896423006751406, 8.280936183075430691237931889684, 9.188361285386150226724310731212, 10.68337238054396286389407782446, 11.28170088562610870466850002921, 11.71699794979689066262019168169

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