Properties

Label 2-675-5.4-c3-0-51
Degree $2$
Conductor $675$
Sign $0.894 + 0.447i$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.258i·2-s + 7.93·4-s + 14.5i·7-s + 4.12i·8-s + 49.2·11-s − 72.1i·13-s − 3.75·14-s + 62.3·16-s − 118. i·17-s − 123.·19-s + 12.7i·22-s − 91.4i·23-s + 18.6·26-s + 115. i·28-s + 174.·29-s + ⋯
L(s)  = 1  + 0.0914i·2-s + 0.991·4-s + 0.783i·7-s + 0.182i·8-s + 1.35·11-s − 1.53i·13-s − 0.0716·14-s + 0.974·16-s − 1.68i·17-s − 1.48·19-s + 0.123i·22-s − 0.829i·23-s + 0.140·26-s + 0.777i·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.775819968\)
\(L(\frac12)\) \(\approx\) \(2.775819968\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 0.258iT - 8T^{2} \)
7 \( 1 - 14.5iT - 343T^{2} \)
11 \( 1 - 49.2T + 1.33e3T^{2} \)
13 \( 1 + 72.1iT - 2.19e3T^{2} \)
17 \( 1 + 118. iT - 4.91e3T^{2} \)
19 \( 1 + 123.T + 6.85e3T^{2} \)
23 \( 1 + 91.4iT - 1.21e4T^{2} \)
29 \( 1 - 174.T + 2.43e4T^{2} \)
31 \( 1 + 46.2T + 2.97e4T^{2} \)
37 \( 1 - 154. iT - 5.06e4T^{2} \)
41 \( 1 - 364.T + 6.89e4T^{2} \)
43 \( 1 + 125. iT - 7.95e4T^{2} \)
47 \( 1 - 221. iT - 1.03e5T^{2} \)
53 \( 1 + 13.6iT - 1.48e5T^{2} \)
59 \( 1 - 239.T + 2.05e5T^{2} \)
61 \( 1 + 54.5T + 2.26e5T^{2} \)
67 \( 1 + 76.0iT - 3.00e5T^{2} \)
71 \( 1 + 728.T + 3.57e5T^{2} \)
73 \( 1 - 501. iT - 3.89e5T^{2} \)
79 \( 1 + 397.T + 4.93e5T^{2} \)
83 \( 1 + 1.36e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.46e3T + 7.04e5T^{2} \)
97 \( 1 - 335. iT - 9.12e5T^{2} \)
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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

−10.16294807774722505265696328435, −9.075860399018865763306617892171, −8.322191091759754522241541587699, −7.29758048640475857949668644319, −6.43261027455390484333895962932, −5.75684666730261384845037905508, −4.56342519994204696079251652982, −3.09548739059331655859451273394, −2.33022616579725847727764820506, −0.828149995391711281771873785559, 1.28575157749916072699506230107, 2.10348371509837615737129583890, 3.78286239579947985653421401020, 4.24728976343177361405627388235, 6.08458722929550626111649704771, 6.56242698272141716308875421911, 7.33264386419012854822471944413, 8.450823095107822087322875618020, 9.322574327266213831475954344028, 10.39449246800823800498689538624

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