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Properties

Label 2-275-11.5-c1-0-3
Degree $2$
Conductor $275$
Sign $-0.800 - 0.599i$
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  + (0.647 + 1.99i)2-s + (1.54 + 1.12i)3-s + (−1.93 + 1.40i)4-s + (−1.23 + 3.81i)6-s + (−2.48 + 1.80i)7-s + (−0.661 − 0.480i)8-s + (0.203 + 0.627i)9-s + (1.86 − 2.74i)11-s − 4.57·12-s + (−0.942 − 2.90i)13-s + (−5.19 − 3.77i)14-s + (−0.947 + 2.91i)16-s + (0.143 − 0.441i)17-s + (−1.11 + 0.812i)18-s + (6.38 + 4.64i)19-s + ⋯
L(s)  = 1  + (0.457 + 1.40i)2-s + (0.893 + 0.649i)3-s + (−0.966 + 0.702i)4-s + (−0.505 + 1.55i)6-s + (−0.937 + 0.681i)7-s + (−0.233 − 0.169i)8-s + (0.0679 + 0.209i)9-s + (0.561 − 0.827i)11-s − 1.31·12-s + (−0.261 − 0.804i)13-s + (−1.38 − 1.00i)14-s + (−0.236 + 0.729i)16-s + (0.0347 − 0.106i)17-s + (−0.263 + 0.191i)18-s + (1.46 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611954 + 1.83938i\)
\(L(\frac12)\) \(\approx\) \(0.611954 + 1.83938i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.647 - 1.99i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.54 - 1.12i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (2.48 - 1.80i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.942 + 2.90i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.143 + 0.441i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.38 - 4.64i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
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 \( 1 + (-1.49 + 1.08i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.56 - 2.59i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 + (2.41 + 1.75i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.29 + 3.98i)T + (-42.8 + 31.1i)T^{2} \)
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 \( 1 - 6.75T + 67T^{2} \)
71 \( 1 + (2.01 - 6.20i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.98 - 5.80i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.57 + 10.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.75 - 8.48i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (4.74 + 14.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.64899657281384858641745418382, −11.39840518685438167127122964631, −9.879794232603767707038277517870, −9.223676011226257414247149403156, −8.308121868928898908856799457357, −7.41919546738569644473627101320, −6.09319748287351302608329292518, −5.52427607170676065108059704069, −3.93957424927048787794875353354, −3.03118753377796157755362928910, 1.46373662610282744122650657398, 2.73981401483663508296916555633, 3.66141903400689684166408717843, 4.86731643464404126852555282794, 6.89395756841047728752306049638, 7.40822538844591646731611789659, 9.126396807783093875379310599762, 9.616718353027791736067253669353, 10.70131665591185566438274422995, 11.68704013561879202867696199157

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