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Properties

Label 2-275-5.3-c2-0-19
Degree $2$
Conductor $275$
Sign $-0.437 + 0.899i$
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  + (0.302 − 0.302i)2-s + (−2.81 − 2.81i)3-s + 3.81i·4-s − 1.70·6-s + (3.80 − 3.80i)7-s + (2.36 + 2.36i)8-s + 6.85i·9-s + 3.31·11-s + (10.7 − 10.7i)12-s + (−7.59 − 7.59i)13-s − 2.30i·14-s − 13.8·16-s + (11.0 − 11.0i)17-s + (2.07 + 2.07i)18-s − 15.5i·19-s + ⋯
L(s)  = 1  + (0.151 − 0.151i)2-s + (−0.938 − 0.938i)3-s + 0.954i·4-s − 0.284·6-s + (0.543 − 0.543i)7-s + (0.295 + 0.295i)8-s + 0.761i·9-s + 0.301·11-s + (0.895 − 0.895i)12-s + (−0.584 − 0.584i)13-s − 0.164i·14-s − 0.864·16-s + (0.651 − 0.651i)17-s + (0.115 + 0.115i)18-s − 0.818i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.437 + 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.543124 - 0.868319i\)
\(L(\frac12)\) \(\approx\) \(0.543124 - 0.868319i\)
\(L(2)\) 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 \( 1 - 3.31T \)
good2 \( 1 + (-0.302 + 0.302i)T - 4iT^{2} \)
3 \( 1 + (2.81 + 2.81i)T + 9iT^{2} \)
7 \( 1 + (-3.80 + 3.80i)T - 49iT^{2} \)
13 \( 1 + (7.59 + 7.59i)T + 169iT^{2} \)
17 \( 1 + (-11.0 + 11.0i)T - 289iT^{2} \)
19 \( 1 + 15.5iT - 361T^{2} \)
23 \( 1 + (18.8 + 18.8i)T + 529iT^{2} \)
29 \( 1 + 45.1iT - 841T^{2} \)
31 \( 1 + 19.2T + 961T^{2} \)
37 \( 1 + (-37.8 + 37.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 21.9T + 1.68e3T^{2} \)
43 \( 1 + (28.4 + 28.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (21.0 - 21.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (24.2 + 24.2i)T + 2.80e3iT^{2} \)
59 \( 1 - 84.9iT - 3.48e3T^{2} \)
61 \( 1 - 104.T + 3.72e3T^{2} \)
67 \( 1 + (-58.0 + 58.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 80.1T + 5.04e3T^{2} \)
73 \( 1 + (-28.1 - 28.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 155. iT - 6.24e3T^{2} \)
83 \( 1 + (7.52 + 7.52i)T + 6.88e3iT^{2} \)
89 \( 1 - 116. iT - 7.92e3T^{2} \)
97 \( 1 + (123. - 123. i)T - 9.40e3iT^{2} \)
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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

−11.59405623168407140569984723565, −10.85251583013834868425634150698, −9.492279675444151236196057667628, −8.017437357251417258489785411321, −7.47476277115835276890051718490, −6.52347969107381017032789404598, −5.24257561278003425964611988614, −4.05226154784529475899890303978, −2.39405214555825421857203022387, −0.56401204896086715978325194713, 1.66860059629791821225774191326, 3.96538107941058420099610170035, 5.08768019007151959255076555654, 5.62627330021628131163763100951, 6.66940262736201945977123046671, 8.180480095591099277169887660072, 9.591931981871178746128858585818, 10.00967689901103394369757360043, 11.09113748126079858857398169675, 11.62885152476914571205854872296

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