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Properties

Label 2-825-15.8-c1-0-58
Degree $2$
Conductor $825$
Sign $-0.927 + 0.374i$
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  + (0.292 + 0.292i)2-s + (0.292 − 1.70i)3-s − 1.82i·4-s + (0.585 − 0.414i)6-s + (−0.585 + 0.585i)7-s + (1.12 − 1.12i)8-s + (−2.82 − i)9-s + i·11-s + (−3.12 − 0.535i)12-s + (−3.41 − 3.41i)13-s − 0.343·14-s − 3·16-s + (−2 − 2i)17-s + (−0.535 − 1.12i)18-s − 0.828i·19-s + ⋯
L(s)  = 1  + (0.207 + 0.207i)2-s + (0.169 − 0.985i)3-s − 0.914i·4-s + (0.239 − 0.169i)6-s + (−0.221 + 0.221i)7-s + (0.396 − 0.396i)8-s + (−0.942 − 0.333i)9-s + 0.301i·11-s + (−0.901 − 0.154i)12-s + (−0.946 − 0.946i)13-s − 0.0917·14-s − 0.750·16-s + (−0.485 − 0.485i)17-s + (−0.126 − 0.264i)18-s − 0.190i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.231842 - 1.19373i\)
\(L(\frac12)\) \(\approx\) \(0.231842 - 1.19373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.292 + 1.70i)T \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-0.292 - 0.292i)T + 2iT^{2} \)
7 \( 1 + (0.585 - 0.585i)T - 7iT^{2} \)
13 \( 1 + (3.41 + 3.41i)T + 13iT^{2} \)
17 \( 1 + (2 + 2i)T + 17iT^{2} \)
19 \( 1 + 0.828iT - 19T^{2} \)
23 \( 1 + (0.828 - 0.828i)T - 23iT^{2} \)
29 \( 1 - 8.82T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (5.65 - 5.65i)T - 37iT^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 + (4.58 + 4.58i)T + 43iT^{2} \)
47 \( 1 + (4.82 + 4.82i)T + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 - 2.34T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-2.58 + 2.58i)T - 67iT^{2} \)
71 \( 1 + 9.65iT - 71T^{2} \)
73 \( 1 + (4.58 + 4.58i)T + 73iT^{2} \)
79 \( 1 - 4.82iT - 79T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (-1.65 + 1.65i)T - 97iT^{2} \)
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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

−9.933028766617673886508738731141, −8.948831905542115031075356904352, −8.055077039462463252707958387670, −7.02345850383876131462580843015, −6.52753769437103760566372414465, −5.47440605728043442488593108616, −4.75923039704264644849448770552, −3.03890962679409388841417329251, −1.99685863304220314870868752202, −0.51881742896170273844016789002, 2.33884090271654022187514507164, 3.28781530722005834460899896745, 4.28444265971986336945688110293, 4.82690081537264511203168103788, 6.24430954658328204813905575526, 7.22669402456778240561470611013, 8.303322542907089073291601075898, 8.837860039100878209017308279484, 9.838382073824225795456215823546, 10.52277750174950606819910869413

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