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Properties

Label 2-275-5.3-c2-0-6
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  + (1.02 − 1.02i)2-s + (−0.314 − 0.314i)3-s + 1.91i·4-s − 0.642·6-s + (−6.75 + 6.75i)7-s + (6.03 + 6.03i)8-s − 8.80i·9-s − 3.31·11-s + (0.603 − 0.603i)12-s + (16.4 + 16.4i)13-s + 13.7i·14-s + 4.66·16-s + (−17.2 + 17.2i)17-s + (−8.98 − 8.98i)18-s + 28.5i·19-s + ⋯
L(s)  = 1  + (0.510 − 0.510i)2-s + (−0.104 − 0.104i)3-s + 0.479i·4-s − 0.107·6-s + (−0.964 + 0.964i)7-s + (0.754 + 0.754i)8-s − 0.977i·9-s − 0.301·11-s + (0.0502 − 0.0502i)12-s + (1.26 + 1.26i)13-s + 0.984i·14-s + 0.291·16-s + (−1.01 + 1.01i)17-s + (−0.499 − 0.499i)18-s + 1.50i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.39456 + 0.872286i\)
\(L(\frac12)\) \(\approx\) \(1.39456 + 0.872286i\)
\(L(2)\) 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 \( 1 + (-1.02 + 1.02i)T - 4iT^{2} \)
3 \( 1 + (0.314 + 0.314i)T + 9iT^{2} \)
7 \( 1 + (6.75 - 6.75i)T - 49iT^{2} \)
13 \( 1 + (-16.4 - 16.4i)T + 169iT^{2} \)
17 \( 1 + (17.2 - 17.2i)T - 289iT^{2} \)
19 \( 1 - 28.5iT - 361T^{2} \)
23 \( 1 + (-5.89 - 5.89i)T + 529iT^{2} \)
29 \( 1 + 43.9iT - 841T^{2} \)
31 \( 1 - 36.3T + 961T^{2} \)
37 \( 1 + (-21.6 + 21.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 34.5T + 1.68e3T^{2} \)
43 \( 1 + (-6.04 - 6.04i)T + 1.84e3iT^{2} \)
47 \( 1 + (12.0 - 12.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (40.1 + 40.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 15.3iT - 3.48e3T^{2} \)
61 \( 1 - 116.T + 3.72e3T^{2} \)
67 \( 1 + (-39.2 + 39.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 53.8T + 5.04e3T^{2} \)
73 \( 1 + (20.3 + 20.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 0.0339iT - 6.24e3T^{2} \)
83 \( 1 + (-10.5 - 10.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 89.4iT - 7.92e3T^{2} \)
97 \( 1 + (-23.9 + 23.9i)T - 9.40e3iT^{2} \)
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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

−11.86968811964746843612404624842, −11.32849140444636850994682856557, −9.972804540462382646795518587703, −8.934512054545658813429546947674, −8.181537697904987357883719749817, −6.53427978422719482489512330651, −5.98666977257399619799426822642, −4.19512245821731317718617559562, −3.41684864317085681853658478148, −2.00393471382856998933721992548, 0.72291030972569340735121934861, 2.99700786817671559250949809015, 4.48401818286408420983943478781, 5.33168050231378256255820770701, 6.56850327850880087380023823892, 7.19391740539429478849810010446, 8.527269423332406212705935661377, 9.828094362519256832032478439791, 10.62911532629764428053306872651, 11.14632902356181366231391852305

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