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Properties

Label 2-675-5.4-c3-0-48
Degree $2$
Conductor $675$
Sign $0.447 + 0.894i$
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  − 1.33i·2-s + 6.23·4-s + 10.6i·7-s − 18.9i·8-s − 11.2·11-s + 2.74i·13-s + 14.2·14-s + 24.6·16-s − 29.5i·17-s + 31.1·19-s + 14.9i·22-s − 116. i·23-s + 3.64·26-s + 66.6i·28-s + 108.·29-s + ⋯
L(s)  = 1  − 0.470i·2-s + 0.778·4-s + 0.577i·7-s − 0.836i·8-s − 0.308·11-s + 0.0584i·13-s + 0.271·14-s + 0.385·16-s − 0.421i·17-s + 0.375·19-s + 0.145i·22-s − 1.06i·23-s + 0.0274·26-s + 0.449i·28-s + 0.694·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
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: \((2,\ 675,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.508361938\)
\(L(\frac12)\) \(\approx\) \(2.508361938\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.33iT - 8T^{2} \)
7 \( 1 - 10.6iT - 343T^{2} \)
11 \( 1 + 11.2T + 1.33e3T^{2} \)
13 \( 1 - 2.74iT - 2.19e3T^{2} \)
17 \( 1 + 29.5iT - 4.91e3T^{2} \)
19 \( 1 - 31.1T + 6.85e3T^{2} \)
23 \( 1 + 116. iT - 1.21e4T^{2} \)
29 \( 1 - 108.T + 2.43e4T^{2} \)
31 \( 1 - 70.7T + 2.97e4T^{2} \)
37 \( 1 + 282. iT - 5.06e4T^{2} \)
41 \( 1 - 425.T + 6.89e4T^{2} \)
43 \( 1 - 312. iT - 7.95e4T^{2} \)
47 \( 1 + 193. iT - 1.03e5T^{2} \)
53 \( 1 + 103. iT - 1.48e5T^{2} \)
59 \( 1 + 494.T + 2.05e5T^{2} \)
61 \( 1 - 424.T + 2.26e5T^{2} \)
67 \( 1 - 586. iT - 3.00e5T^{2} \)
71 \( 1 - 1.13e3T + 3.57e5T^{2} \)
73 \( 1 - 302. iT - 3.89e5T^{2} \)
79 \( 1 - 525.T + 4.93e5T^{2} \)
83 \( 1 + 1.00e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + 25.6iT - 9.12e5T^{2} \)
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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

−10.07484581088328910224346238350, −9.240272807814901432285681725073, −8.206143971799723023695729274089, −7.27618896897927785374705884422, −6.38985127283398182423419388338, −5.49458873267096396996405467189, −4.25028486979776836302509820517, −2.94341976999188668136600676289, −2.23492630114841871240142166178, −0.78652694254918412319672771927, 1.14184573614934930963065933098, 2.48541209100856890094914441112, 3.63304528623083274486311925090, 4.94191943131129116557416728096, 5.91042751556283233837504246266, 6.75623747296402498718429125067, 7.60566059893436210619276893020, 8.198516149999966514618879318158, 9.414385524004163624117304852792, 10.38148564996123087497161129043

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