Properties

Label 2-275-5.4-c5-0-18
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  − 7.82i·2-s − 16.3i·3-s − 29.2·4-s − 127.·6-s + 125. i·7-s − 21.7i·8-s − 22.8·9-s − 121·11-s + 476. i·12-s + 532. i·13-s + 981.·14-s − 1.10e3·16-s + 1.37e3i·17-s + 178. i·18-s + 554.·19-s + ⋯
L(s)  = 1  − 1.38i·2-s − 1.04i·3-s − 0.913·4-s − 1.44·6-s + 0.967i·7-s − 0.119i·8-s − 0.0939·9-s − 0.301·11-s + 0.955i·12-s + 0.873i·13-s + 1.33·14-s − 1.07·16-s + 1.15i·17-s + 0.129i·18-s + 0.352·19-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

\(L(3)\) \(\approx\) \(1.729312350\)
\(L(\frac12)\) \(\approx\) \(1.729312350\)
\(L(\frac{7}{2})\) 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 \( 1 + 7.82iT - 32T^{2} \)
3 \( 1 + 16.3iT - 243T^{2} \)
7 \( 1 - 125. iT - 1.68e4T^{2} \)
13 \( 1 - 532. iT - 3.71e5T^{2} \)
17 \( 1 - 1.37e3iT - 1.41e6T^{2} \)
19 \( 1 - 554.T + 2.47e6T^{2} \)
23 \( 1 - 4.25e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.97e3T + 2.05e7T^{2} \)
31 \( 1 - 3.13e3T + 2.86e7T^{2} \)
37 \( 1 + 1.38e3iT - 6.93e7T^{2} \)
41 \( 1 - 679.T + 1.15e8T^{2} \)
43 \( 1 + 1.72e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.51e4iT - 2.29e8T^{2} \)
53 \( 1 + 9.54e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.75e4T + 7.14e8T^{2} \)
61 \( 1 + 4.05e4T + 8.44e8T^{2} \)
67 \( 1 - 5.87e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.25e4T + 1.80e9T^{2} \)
73 \( 1 - 2.37e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.86e4T + 3.07e9T^{2} \)
83 \( 1 - 5.22e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.15e3T + 5.58e9T^{2} \)
97 \( 1 + 7.90e4iT - 8.58e9T^{2} \)
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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

−11.14025087337468440883843962468, −10.05940159722374300477844036039, −9.172037111815892633385546560936, −8.104905470571102696857045757593, −6.92496945677798037463272092820, −5.90572054534792775424902605572, −4.35861960538409792697316416380, −2.98395063810949489420135424732, −1.94973779038580888281699121178, −1.18804086115169125224109821918, 0.52300621238767323444565349515, 2.96533966524827307486029696744, 4.49261166656144239728868714023, 4.99522905341780990209642273741, 6.32376191259859267602503435156, 7.27270762294051806298966315333, 8.113506588321672192437855820714, 9.167055682923810452871810236067, 10.26116416057264228147739505143, 10.76488528295698019246850380417

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