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Properties

Label 2-275-5.4-c5-0-34
Degree $2$
Conductor $275$
Sign $-0.447 + 0.894i$
Analytic cond. $44.1055$
Root an. cond. $6.64120$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.5i·2-s + 10.1i·3-s − 78.6·4-s + 106.·6-s − 228. i·7-s + 490. i·8-s + 140.·9-s + 121·11-s − 798. i·12-s + 640. i·13-s − 2.39e3·14-s + 2.64e3·16-s + 2.08e3i·17-s − 1.47e3i·18-s + 2.33e3·19-s + ⋯
L(s)  = 1  − 1.85i·2-s + 0.651i·3-s − 2.45·4-s + 1.21·6-s − 1.75i·7-s + 2.70i·8-s + 0.576·9-s + 0.301·11-s − 1.59i·12-s + 1.05i·13-s − 3.27·14-s + 2.58·16-s + 1.74i·17-s − 1.07i·18-s + 1.48·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(44.1055\)
Root analytic conductor: \(6.64120\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :5/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.881617866\)
\(L(\frac12)\) \(\approx\) \(1.881617866\)
\(L(\frac{7}{2})\) 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 \( 1 - 121T \)
good2 \( 1 + 10.5iT - 32T^{2} \)
3 \( 1 - 10.1iT - 243T^{2} \)
7 \( 1 + 228. iT - 1.68e4T^{2} \)
13 \( 1 - 640. iT - 3.71e5T^{2} \)
17 \( 1 - 2.08e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.33e3T + 2.47e6T^{2} \)
23 \( 1 + 570. iT - 6.43e6T^{2} \)
29 \( 1 - 178.T + 2.05e7T^{2} \)
31 \( 1 - 6.49e3T + 2.86e7T^{2} \)
37 \( 1 + 406. iT - 6.93e7T^{2} \)
41 \( 1 + 1.52e3T + 1.15e8T^{2} \)
43 \( 1 + 2.52e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.05e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.26e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.65e3T + 7.14e8T^{2} \)
61 \( 1 - 6.71e3T + 8.44e8T^{2} \)
67 \( 1 + 9.19e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.87e4T + 1.80e9T^{2} \)
73 \( 1 - 1.30e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.00e5T + 3.07e9T^{2} \)
83 \( 1 - 2.10e4iT - 3.93e9T^{2} \)
89 \( 1 - 2.75e4T + 5.58e9T^{2} \)
97 \( 1 - 9.05e4iT - 8.58e9T^{2} \)
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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

−10.62987829600254942637125897795, −10.08254411155209498327949565209, −9.458979776740635578588017634861, −8.213108293106262310930836963619, −6.84647771076003726429742348050, −4.87062059127641591211196277534, −3.99436367647046081032120707974, −3.58382820903257235450721548377, −1.72267244196284062309727157595, −0.842264891715174175222154475171, 0.831636359775579765073359536972, 2.89053605389714983146751361161, 4.83179066444660044070442051207, 5.55935734601446557410438921035, 6.43368639257421176384314891738, 7.43790622856893121934645978715, 8.079252334247758997318587698192, 9.175805643635557321438862077823, 9.727986812946658863577776453470, 11.76889007164311348214512086367

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