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Properties

Label 2-825-11.3-c1-0-23
Degree $2$
Conductor $825$
Sign $0.751 + 0.659i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (0.190 − 0.587i)4-s + (−1.30 − 0.951i)6-s + (−0.927 + 2.85i)7-s + (0.690 + 2.12i)8-s + (−0.809 + 0.587i)9-s + (2.80 − 1.76i)11-s − 0.618·12-s + (5.04 − 3.66i)13-s + (1.5 + 4.61i)14-s + (3.92 + 2.85i)16-s + (−0.5 − 0.363i)17-s + (−0.499 + 1.53i)18-s + (−0.263 − 0.812i)19-s + ⋯
L(s)  = 1  + (0.925 − 0.672i)2-s + (−0.178 − 0.549i)3-s + (0.0954 − 0.293i)4-s + (−0.534 − 0.388i)6-s + (−0.350 + 1.07i)7-s + (0.244 + 0.751i)8-s + (−0.269 + 0.195i)9-s + (0.846 − 0.531i)11-s − 0.178·12-s + (1.39 − 1.01i)13-s + (0.400 + 1.23i)14-s + (0.981 + 0.713i)16-s + (−0.121 − 0.0881i)17-s + (−0.117 + 0.362i)18-s + (−0.0605 − 0.186i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33462 - 0.878545i\)
\(L(\frac12)\) \(\approx\) \(2.33462 - 0.878545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
11 \( 1 + (-2.80 + 1.76i)T \)
good2 \( 1 + (-1.30 + 0.951i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.927 - 2.85i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-5.04 + 3.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.47T + 23T^{2} \)
29 \( 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.11 + 2.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.30 + 4.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.83 - 5.65i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + (-0.190 - 0.587i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.97 - 4.33i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.64 - 5.06i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.927 + 0.673i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + (11.7 + 8.55i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.381 - 1.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.427 - 0.310i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.2 + 7.46i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.47T + 89T^{2} \)
97 \( 1 + (-12.1 + 8.83i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.55819877588337139968547867257, −9.032693007749821987858030739047, −8.630061387076913033752634488635, −7.56066283009945787068729771189, −6.17364899300002192912197226579, −5.81575795181926125409648590146, −4.69045219401515244383716389448, −3.40461081004274588252449003868, −2.80195510001472176980855940453, −1.35410848970383396818218321580, 1.23861158627981513011651957616, 3.49920101415938192584510478330, 4.12260353507927922030783595375, 4.78697101678138887233239429765, 6.06700034358211929452775101708, 6.61693588325579851170971818714, 7.33283314748248424864949418338, 8.705333797522664601867646425887, 9.549902124518637130489554638271, 10.30791039584508974325746558313

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