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Properties

Label 2-1472-16.13-c1-0-40
Degree $2$
Conductor $1472$
Sign $-0.694 + 0.719i$
Analytic cond. $11.7539$
Root an. cond. $3.42840$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.339 − 0.339i)3-s + (1.46 − 1.46i)5-s − 2.63i·7-s − 2.76i·9-s + (2.99 − 2.99i)11-s + (−0.749 − 0.749i)13-s − 0.998·15-s − 3.64·17-s + (−1.76 − 1.76i)19-s + (−0.893 + 0.893i)21-s + i·23-s + 0.682i·25-s + (−1.95 + 1.95i)27-s + (0.0790 + 0.0790i)29-s + 2.07·31-s + ⋯
L(s)  = 1  + (−0.196 − 0.196i)3-s + (0.657 − 0.657i)5-s − 0.994i·7-s − 0.923i·9-s + (0.901 − 0.901i)11-s + (−0.207 − 0.207i)13-s − 0.257·15-s − 0.885·17-s + (−0.405 − 0.405i)19-s + (−0.195 + 0.195i)21-s + 0.208i·23-s + 0.136i·25-s + (−0.377 + 0.377i)27-s + (0.0146 + 0.0146i)29-s + 0.372·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(11.7539\)
Root analytic conductor: \(3.42840\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (1105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :1/2),\ -0.694 + 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.531484730\)
\(L(\frac12)\) \(\approx\) \(1.531484730\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 - iT \)
good3 \( 1 + (0.339 + 0.339i)T + 3iT^{2} \)
5 \( 1 + (-1.46 + 1.46i)T - 5iT^{2} \)
7 \( 1 + 2.63iT - 7T^{2} \)
11 \( 1 + (-2.99 + 2.99i)T - 11iT^{2} \)
13 \( 1 + (0.749 + 0.749i)T + 13iT^{2} \)
17 \( 1 + 3.64T + 17T^{2} \)
19 \( 1 + (1.76 + 1.76i)T + 19iT^{2} \)
29 \( 1 + (-0.0790 - 0.0790i)T + 29iT^{2} \)
31 \( 1 - 2.07T + 31T^{2} \)
37 \( 1 + (3.64 - 3.64i)T - 37iT^{2} \)
41 \( 1 - 1.90iT - 41T^{2} \)
43 \( 1 + (6.84 - 6.84i)T - 43iT^{2} \)
47 \( 1 - 8.55T + 47T^{2} \)
53 \( 1 + (6.62 - 6.62i)T - 53iT^{2} \)
59 \( 1 + (-5.45 + 5.45i)T - 59iT^{2} \)
61 \( 1 + (-2.13 - 2.13i)T + 61iT^{2} \)
67 \( 1 + (3.51 + 3.51i)T + 67iT^{2} \)
71 \( 1 - 1.61iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + (1.26 + 1.26i)T + 83iT^{2} \)
89 \( 1 + 12.3iT - 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−9.136048999898600361553651747593, −8.647850105369341503831675214073, −7.50664723562352460004876338054, −6.56152091720335596977640825455, −6.14981092234334337336735138871, −5.00248054576824184220464406610, −4.11158516940496739591226015037, −3.18166786802066162672721783911, −1.57388281266249956243126653540, −0.62348149970286958057789895013, 2.00972565236967846728183443209, 2.39066664833101758979989193297, 3.93138770126395658925667866546, 4.87252417226032992611450445880, 5.71185550177479787483435901757, 6.55182171454828723040564485163, 7.17726576265476309452070821562, 8.373995017107906688388914553260, 9.034749965875839601890038462067, 9.901006615499911310939795627857

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