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Properties

Label 2-675-5.4-c3-0-23
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

\(L(2)\) \(\approx\) \(2.307271821\)
\(L(\frac12)\) \(\approx\) \(2.307271821\)
\(L(\frac{5}{2})\) 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

−10.40700032959589626512057386728, −9.436285544308417357805092712704, −8.528551108183198074523288695373, −7.65483156477285179702503446748, −6.86257333996700769711519311031, −5.92209010103927178512629087455, −5.26014557971315508913978936160, −3.77895394555051968451626645806, −2.59588789299597101455863571288, −1.47622351081247531649209860846, 0.62822123611797604599794552157, 1.85638622800645088692315453499, 3.03845136173466483653129007399, 3.98675213492286921663736740609, 5.21809971052963330803243658522, 6.47577596144935525232985940515, 7.04665182355310043631466913961, 8.008989977468248714245274279080, 9.082693949374629117432773296753, 10.18896780827195859290174541383

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