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Properties

Label 2-275-11.5-c1-0-11
Degree $2$
Conductor $275$
Sign $0.501 + 0.865i$
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  + (−0.147 − 0.453i)2-s + (0.261 + 0.189i)3-s + (1.43 − 1.04i)4-s + (0.0476 − 0.146i)6-s + (2.17 − 1.57i)7-s + (−1.45 − 1.05i)8-s + (−0.894 − 2.75i)9-s + (−2.79 + 1.79i)11-s + 0.572·12-s + (1.44 + 4.43i)13-s + (−1.03 − 0.753i)14-s + (0.829 − 2.55i)16-s + (1.42 − 4.39i)17-s + (−1.11 + 0.812i)18-s + (3.51 + 2.55i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.320i)2-s + (0.150 + 0.109i)3-s + (0.716 − 0.520i)4-s + (0.0194 − 0.0598i)6-s + (0.821 − 0.596i)7-s + (−0.514 − 0.374i)8-s + (−0.298 − 0.917i)9-s + (−0.841 + 0.540i)11-s + 0.165·12-s + (0.400 + 1.23i)13-s + (−0.277 − 0.201i)14-s + (0.207 − 0.638i)16-s + (0.346 − 1.06i)17-s + (−0.263 + 0.191i)18-s + (0.805 + 0.585i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.501 + 0.865i$
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: \((2,\ 275,\ (\ :1/2),\ 0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29881 - 0.748426i\)
\(L(\frac12)\) \(\approx\) \(1.29881 - 0.748426i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.79 - 1.79i)T \)
good2 \( 1 + (0.147 + 0.453i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.261 - 0.189i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-2.17 + 1.57i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.44 - 4.43i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.42 + 4.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.51 - 2.55i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 + (-2.43 + 1.77i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.737 - 2.26i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.61 - 6.25i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.78 - 1.29i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.06T + 43T^{2} \)
47 \( 1 + (-3.52 - 2.56i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.95 - 6.02i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.50 - 6.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.23 - 3.78i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.31T + 67T^{2} \)
71 \( 1 + (-0.369 + 1.13i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.826 + 0.600i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.08 - 3.33i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.43 - 10.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.76T + 89T^{2} \)
97 \( 1 + (5.72 + 17.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.78742589273530725213312599169, −10.78896458941323293200000374756, −9.952209506093329418187123001704, −9.097877743103058326290356859326, −7.72675712030551526899461668311, −6.87291421843585941459897696698, −5.70057585234838155754562667366, −4.40228473625348388030040210539, −2.91173594800380146361376987384, −1.37688925731274328176941569321, 2.16071957952268526343087384355, 3.28847953488222673301242315659, 5.24998739883528098723687463103, 5.90991978373996097201130960487, 7.51578256670309026727829441500, 8.080703876771650562820833342623, 8.707033005091074022528167942020, 10.53198367443596761626199728010, 10.98067649557598124269878674353, 12.04015628652539157185304615986

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