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Properties

Label 2-275-55.29-c2-0-25
Degree $2$
Conductor $275$
Sign $0.230 + 0.972i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.48 + 1.80i)2-s + (−3.80 − 1.23i)3-s + (1.69 + 5.20i)4-s + (−7.23 − 9.95i)6-s + (−2.48 − 7.66i)7-s + (−1.40 + 4.30i)8-s + (5.66 + 4.11i)9-s + (−10.8 + 1.67i)11-s − 21.8i·12-s + (−11.3 − 8.25i)13-s + (7.66 − 23.5i)14-s + (6.42 − 4.66i)16-s + (19.6 − 14.2i)17-s + (6.65 + 20.4i)18-s + (−8.02 − 2.60i)19-s + ⋯
L(s)  = 1  + (1.24 + 0.904i)2-s + (−1.26 − 0.412i)3-s + (0.422 + 1.30i)4-s + (−1.20 − 1.65i)6-s + (−0.355 − 1.09i)7-s + (−0.175 + 0.538i)8-s + (0.629 + 0.457i)9-s + (−0.988 + 0.152i)11-s − 1.82i·12-s + (−0.873 − 0.634i)13-s + (0.547 − 1.68i)14-s + (0.401 − 0.291i)16-s + (1.15 − 0.839i)17-s + (0.369 + 1.13i)18-s + (−0.422 − 0.137i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.230 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.230 + 0.972i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ 0.230 + 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.914516 - 0.722845i\)
\(L(\frac12)\) \(\approx\) \(0.914516 - 0.722845i\)
\(L(2)\) 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 \( 1 + (10.8 - 1.67i)T \)
good2 \( 1 + (-2.48 - 1.80i)T + (1.23 + 3.80i)T^{2} \)
3 \( 1 + (3.80 + 1.23i)T + (7.28 + 5.29i)T^{2} \)
7 \( 1 + (2.48 + 7.66i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (11.3 + 8.25i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-19.6 + 14.2i)T + (89.3 - 274. i)T^{2} \)
19 \( 1 + (8.02 + 2.60i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 25.3iT - 529T^{2} \)
29 \( 1 + (34.7 - 11.3i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-8.38 - 6.08i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (19.2 - 6.26i)T + (1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-53.1 - 17.2i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 37.2T + 1.84e3T^{2} \)
47 \( 1 + (81.3 + 26.4i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (32.0 - 44.1i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (18.3 + 56.3i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-21.5 - 29.7i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 17.8iT - 4.48e3T^{2} \)
71 \( 1 + (4.70 - 3.42i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-15.8 - 48.6i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (20.1 - 27.7i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-117. + 85.7i)T + (2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 127.T + 7.92e3T^{2} \)
97 \( 1 + (-53.0 + 73.0i)T + (-2.90e3 - 8.94e3i)T^{2} \)
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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

−11.83273649459378326749442172669, −10.65837770394592611965164212820, −9.938104678240778159277754150518, −7.80764322791097486228085824656, −7.18587593644272578272561422172, −6.35132840280352693817811112515, −5.31444175567294275228480810097, −4.73675640659786650790508332625, −3.20650561451757466431989753596, −0.43711175069851998009294121244, 2.12150024876436976546718185274, 3.47791680085500405329234572268, 4.83154810465389661382980285063, 5.52635094350493804785132033754, 6.14718775972624426647711872891, 7.910064797137352686713310287414, 9.547555170260467992615995719614, 10.38977440035234956371513727029, 11.25365885118844922524939086315, 11.93776344882074927780374176785

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