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Properties

Label 2-675-5.4-c3-0-37
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  + 2.21i·2-s + 3.10·4-s − 17.1i·7-s + 24.5i·8-s + 66.8·11-s + 72.7i·13-s + 37.8·14-s − 29.5·16-s − 40.2i·17-s − 38.4·19-s + 147. i·22-s − 204. i·23-s − 160.·26-s − 53.1i·28-s + 21.6·29-s + ⋯
L(s)  = 1  + 0.782i·2-s + 0.388·4-s − 0.923i·7-s + 1.08i·8-s + 1.83·11-s + 1.55i·13-s + 0.722·14-s − 0.461·16-s − 0.574i·17-s − 0.464·19-s + 1.43i·22-s − 1.85i·23-s − 1.21·26-s − 0.358i·28-s + 0.138·29-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

\(L(2)\) \(\approx\) \(2.750773177\)
\(L(\frac12)\) \(\approx\) \(2.750773177\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.21iT - 8T^{2} \)
7 \( 1 + 17.1iT - 343T^{2} \)
11 \( 1 - 66.8T + 1.33e3T^{2} \)
13 \( 1 - 72.7iT - 2.19e3T^{2} \)
17 \( 1 + 40.2iT - 4.91e3T^{2} \)
19 \( 1 + 38.4T + 6.85e3T^{2} \)
23 \( 1 + 204. iT - 1.21e4T^{2} \)
29 \( 1 - 21.6T + 2.43e4T^{2} \)
31 \( 1 - 128.T + 2.97e4T^{2} \)
37 \( 1 + 19.4iT - 5.06e4T^{2} \)
41 \( 1 - 270.T + 6.89e4T^{2} \)
43 \( 1 - 242. iT - 7.95e4T^{2} \)
47 \( 1 - 307. iT - 1.03e5T^{2} \)
53 \( 1 + 289. iT - 1.48e5T^{2} \)
59 \( 1 + 17.7T + 2.05e5T^{2} \)
61 \( 1 - 764.T + 2.26e5T^{2} \)
67 \( 1 - 532. iT - 3.00e5T^{2} \)
71 \( 1 - 409.T + 3.57e5T^{2} \)
73 \( 1 - 220. iT - 3.89e5T^{2} \)
79 \( 1 + 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 253. iT - 5.71e5T^{2} \)
89 \( 1 - 1.62e3T + 7.04e5T^{2} \)
97 \( 1 + 457. iT - 9.12e5T^{2} \)
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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

−10.21971436849830803686019532688, −9.185230999338743256753445585413, −8.484447639096318971481577927955, −7.33222505211876497024655155334, −6.61183686604067481860489688717, −6.30818060768700557381143357087, −4.64569871011941811598968709957, −4.01430994806168239296842725442, −2.37827406634467403374188939911, −1.08611007071265863810287894762, 0.962203306531801599169943053161, 2.02108461558097542529930125148, 3.19354091849253767086689690512, 3.98059594306221659786672683435, 5.55960090054624009449223949134, 6.26823578694765027889781461481, 7.27425914925442840956591180291, 8.385202995349120284692981295610, 9.281204630491198847548096318701, 10.01232119303422154247860308822

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