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Properties

Label 2-275-55.9-c1-0-8
Degree $2$
Conductor $275$
Sign $0.638 - 0.769i$
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  + (2.14 + 0.697i)2-s + (0.456 + 0.628i)3-s + (2.50 + 1.82i)4-s + (0.542 + 1.66i)6-s + (0.0728 − 0.100i)7-s + (1.45 + 2.00i)8-s + (0.740 − 2.27i)9-s + (−0.899 + 3.19i)11-s + 2.40i·12-s + (−5.22 − 1.69i)13-s + (0.226 − 0.164i)14-s + (−0.184 − 0.566i)16-s + (−0.494 + 0.160i)17-s + (3.17 − 4.37i)18-s + (2.55 − 1.85i)19-s + ⋯
L(s)  = 1  + (1.51 + 0.493i)2-s + (0.263 + 0.363i)3-s + (1.25 + 0.910i)4-s + (0.221 + 0.681i)6-s + (0.0275 − 0.0379i)7-s + (0.515 + 0.709i)8-s + (0.246 − 0.759i)9-s + (−0.271 + 0.962i)11-s + 0.695i·12-s + (−1.44 − 0.470i)13-s + (0.0605 − 0.0439i)14-s + (−0.0460 − 0.141i)16-s + (−0.119 + 0.0389i)17-s + (0.749 − 1.03i)18-s + (0.586 − 0.426i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.638 - 0.769i$
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: \((2,\ 275,\ (\ :1/2),\ 0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56379 + 1.20415i\)
\(L(\frac12)\) \(\approx\) \(2.56379 + 1.20415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.899 - 3.19i)T \)
good2 \( 1 + (-2.14 - 0.697i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.456 - 0.628i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.0728 + 0.100i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (5.22 + 1.69i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.494 - 0.160i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.55 + 1.85i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.92iT - 23T^{2} \)
29 \( 1 + (3.29 + 2.39i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.17 + 6.70i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.16 - 7.10i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.10 + 4.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.42iT - 43T^{2} \)
47 \( 1 + (0.268 + 0.369i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.0337 + 0.0109i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.42 + 3.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.37 - 7.31i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.53iT - 67T^{2} \)
71 \( 1 + (-3.79 - 11.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.00 - 6.89i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.93 - 5.96i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.595 + 0.193i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + (1.74 + 0.567i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.26490109476514606463995647787, −11.57045268710277449887002641618, −9.972302581319990857890701385757, −9.405893048411090191836201298294, −7.61303950821314041122867152325, −7.03957773351658983935635193322, −5.70759992912661785694171868784, −4.80922535518057966467451604450, −3.85609791949038708259955451003, −2.64958848022132158129441400913, 2.10853484047711059262748820030, 3.12284371616858382059149000934, 4.56061409234942255466668529845, 5.32292726030485381688678435544, 6.56310684647914183193022472956, 7.65170819537808600131599516699, 8.837911987003485431479740288478, 10.32063986562137791305806481067, 11.04654045289082477817970926731, 12.14241772851044531525377041570

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