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Properties

Label 2-275-5.4-c3-0-14
Degree $2$
Conductor $275$
Sign $-0.447 - 0.894i$
Analytic cond. $16.2255$
Root an. cond. $4.02809$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.56i·2-s − 3.56i·3-s − 22.9·4-s + 19.8·6-s − 6.05i·7-s − 83.0i·8-s + 14.3·9-s − 11·11-s + 81.6i·12-s − 4.38i·13-s + 33.6·14-s + 278.·16-s + 110. i·17-s + 79.6i·18-s + 94.2·19-s + ⋯
L(s)  = 1  + 1.96i·2-s − 0.685i·3-s − 2.86·4-s + 1.34·6-s − 0.326i·7-s − 3.66i·8-s + 0.530·9-s − 0.301·11-s + 1.96i·12-s − 0.0935i·13-s + 0.642·14-s + 4.34·16-s + 1.57i·17-s + 1.04i·18-s + 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(16.2255\)
Root analytic conductor: \(4.02809\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.469159886\)
\(L(\frac12)\) \(\approx\) \(1.469159886\)
\(L(\frac{5}{2})\) 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 \( 1 + 11T \)
good2 \( 1 - 5.56iT - 8T^{2} \)
3 \( 1 + 3.56iT - 27T^{2} \)
7 \( 1 + 6.05iT - 343T^{2} \)
13 \( 1 + 4.38iT - 2.19e3T^{2} \)
17 \( 1 - 110. iT - 4.91e3T^{2} \)
19 \( 1 - 94.2T + 6.85e3T^{2} \)
23 \( 1 - 15.7iT - 1.21e4T^{2} \)
29 \( 1 - 256.T + 2.43e4T^{2} \)
31 \( 1 + 170.T + 2.97e4T^{2} \)
37 \( 1 - 190. iT - 5.06e4T^{2} \)
41 \( 1 - 249.T + 6.89e4T^{2} \)
43 \( 1 - 291. iT - 7.95e4T^{2} \)
47 \( 1 + 182. iT - 1.03e5T^{2} \)
53 \( 1 + 289. iT - 1.48e5T^{2} \)
59 \( 1 + 282.T + 2.05e5T^{2} \)
61 \( 1 - 167.T + 2.26e5T^{2} \)
67 \( 1 - 176. iT - 3.00e5T^{2} \)
71 \( 1 - 919.T + 3.57e5T^{2} \)
73 \( 1 - 154. iT - 3.89e5T^{2} \)
79 \( 1 - 882.T + 4.93e5T^{2} \)
83 \( 1 - 277. iT - 5.71e5T^{2} \)
89 \( 1 - 977.T + 7.04e5T^{2} \)
97 \( 1 - 1.10e3iT - 9.12e5T^{2} \)
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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

−12.30365211915230958156200862246, −10.41760033954673995833860973428, −9.536114399029200935513739095797, −8.316047063313988640208429015931, −7.73216574883631815875090944214, −6.85581944849055318227905035804, −6.08810398602743430818688444520, −4.96016855131287772733825271484, −3.80884362878297945976619025775, −1.00029397773553886878634320796, 0.818520054576956611350462280492, 2.41681227933541063152020160258, 3.46382616359390149130769786299, 4.60183254571979644935984580149, 5.34444877455118688922572436657, 7.56588585431064936082739177274, 9.008953571912181824933498590885, 9.475432262649928472834336641997, 10.33821796048451263676251227891, 11.06684119445269040153759689065

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