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Properties

Label 2-825-11.3-c1-0-22
Degree $2$
Conductor $825$
Sign $0.584 + 0.811i$
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  + (−0.649 + 0.472i)2-s + (−0.309 − 0.951i)3-s + (−0.418 + 1.28i)4-s + (0.649 + 0.472i)6-s + (−0.157 + 0.483i)7-s + (−0.832 − 2.56i)8-s + (−0.809 + 0.587i)9-s + (−3.13 + 1.08i)11-s + 1.35·12-s + (4.15 − 3.02i)13-s + (−0.126 − 0.388i)14-s + (−0.440 − 0.319i)16-s + (−2.70 − 1.96i)17-s + (0.248 − 0.764i)18-s + (−0.362 − 1.11i)19-s + ⋯
L(s)  = 1  + (−0.459 + 0.333i)2-s + (−0.178 − 0.549i)3-s + (−0.209 + 0.644i)4-s + (0.265 + 0.192i)6-s + (−0.0593 + 0.182i)7-s + (−0.294 − 0.906i)8-s + (−0.269 + 0.195i)9-s + (−0.944 + 0.327i)11-s + 0.391·12-s + (1.15 − 0.837i)13-s + (−0.0337 − 0.103i)14-s + (−0.110 − 0.0799i)16-s + (−0.656 − 0.476i)17-s + (0.0585 − 0.180i)18-s + (−0.0831 − 0.255i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.584 + 0.811i$
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: \((2,\ 825,\ (\ :1/2),\ 0.584 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.691137 - 0.353877i\)
\(L(\frac12)\) \(\approx\) \(0.691137 - 0.353877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.13 - 1.08i)T \)
good2 \( 1 + (0.649 - 0.472i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.157 - 0.483i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.15 + 3.02i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.70 + 1.96i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.362 + 1.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.91T + 23T^{2} \)
29 \( 1 + (0.121 - 0.373i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.14 + 3.73i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.37 + 4.23i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.864 + 2.66i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.05T + 43T^{2} \)
47 \( 1 + (3.55 + 10.9i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.93 + 5.76i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.935 - 2.88i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.853 + 0.620i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.31T + 67T^{2} \)
71 \( 1 + (3.31 + 2.40i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.05 + 12.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-13.0 + 9.48i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.4 + 9.02i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.84T + 89T^{2} \)
97 \( 1 + (12.2 - 8.91i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.04096338605740729445015995793, −8.944362458980593761876245476316, −8.371535406252304084662312869814, −7.58605984797816175630065042169, −6.85259568396000122279973072138, −5.88785407696719196420248429389, −4.81428714802041230137608557948, −3.51126690203798566805170074063, −2.43891052982390749755737514312, −0.51589316287518500515780340863, 1.23579218243354840676404394290, 2.70073266290656802558130656102, 4.06097976607716275895043698783, 4.99336207917614760062600827154, 5.93099606047337753867645969412, 6.71851561976830170237521620453, 8.263671264701317304268140573399, 8.708585290716481376058999430683, 9.665876711526610605704403450497, 10.34971027049770672134126494116

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