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Properties

Label 2-825-5.4-c3-0-37
Degree $2$
Conductor $825$
Sign $0.447 - 0.894i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
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  + 1.98i·2-s − 3i·3-s + 4.06·4-s + 5.95·6-s − 5.59i·7-s + 23.9i·8-s − 9·9-s − 11·11-s − 12.1i·12-s − 47.5i·13-s + 11.1·14-s − 15.0·16-s + 66.9i·17-s − 17.8i·18-s − 29.0·19-s + ⋯
L(s)  = 1  + 0.701i·2-s − 0.577i·3-s + 0.507·4-s + 0.405·6-s − 0.302i·7-s + 1.05i·8-s − 0.333·9-s − 0.301·11-s − 0.293i·12-s − 1.01i·13-s + 0.212·14-s − 0.234·16-s + 0.955i·17-s − 0.233i·18-s − 0.351·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.315583146\)
\(L(\frac12)\) \(\approx\) \(2.315583146\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 - 1.98iT - 8T^{2} \)
7 \( 1 + 5.59iT - 343T^{2} \)
13 \( 1 + 47.5iT - 2.19e3T^{2} \)
17 \( 1 - 66.9iT - 4.91e3T^{2} \)
19 \( 1 + 29.0T + 6.85e3T^{2} \)
23 \( 1 - 195. iT - 1.21e4T^{2} \)
29 \( 1 - 273.T + 2.43e4T^{2} \)
31 \( 1 - 156.T + 2.97e4T^{2} \)
37 \( 1 + 185. iT - 5.06e4T^{2} \)
41 \( 1 + 53.8T + 6.89e4T^{2} \)
43 \( 1 - 553. iT - 7.95e4T^{2} \)
47 \( 1 + 581. iT - 1.03e5T^{2} \)
53 \( 1 - 534. iT - 1.48e5T^{2} \)
59 \( 1 - 161.T + 2.05e5T^{2} \)
61 \( 1 + 168.T + 2.26e5T^{2} \)
67 \( 1 - 474. iT - 3.00e5T^{2} \)
71 \( 1 - 422.T + 3.57e5T^{2} \)
73 \( 1 + 1.11e3iT - 3.89e5T^{2} \)
79 \( 1 - 900.T + 4.93e5T^{2} \)
83 \( 1 + 395. iT - 5.71e5T^{2} \)
89 \( 1 - 1.00e3T + 7.04e5T^{2} \)
97 \( 1 - 21.5iT - 9.12e5T^{2} \)
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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

−10.16639267961201852572935792305, −8.787957933451402563745268027701, −7.944177156050381670237275378951, −7.53363744099383091039582250468, −6.48182695438216960220151858937, −5.88526771730277598977306044830, −4.92866787023654248912815089611, −3.41369526764896585618998787156, −2.33447198552694604984182092484, −1.06937214753114643561867529520, 0.68335878329333476521712696829, 2.23939575708996520976284296232, 2.91341268565057960777699169278, 4.17162461321099214601780727724, 4.96239106830765272805042910447, 6.35641050193270021879229140527, 6.87241167433262664077072743208, 8.203804286726686878132536479764, 9.012749152679944992821977666520, 9.955333675412167456464569019186

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