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Properties

Label 2-825-15.8-c1-0-26
Degree $2$
Conductor $825$
Sign $-0.662 - 0.749i$
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  + (1.70 + 1.70i)2-s + (1.70 − 0.292i)3-s + 3.82i·4-s + (3.41 + 2.41i)6-s + (−3.41 + 3.41i)7-s + (−3.12 + 3.12i)8-s + (2.82 − i)9-s + i·11-s + (1.12 + 6.53i)12-s + (−0.585 − 0.585i)13-s − 11.6·14-s − 2.99·16-s + (−2 − 2i)17-s + (6.53 + 3.12i)18-s + 4.82i·19-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)2-s + (0.985 − 0.169i)3-s + 1.91i·4-s + (1.39 + 0.985i)6-s + (−1.29 + 1.29i)7-s + (−1.10 + 1.10i)8-s + (0.942 − 0.333i)9-s + 0.301i·11-s + (0.323 + 1.88i)12-s + (−0.162 − 0.162i)13-s − 3.11·14-s − 0.749·16-s + (−0.485 − 0.485i)17-s + (1.54 + 0.735i)18-s + 1.10i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.662 - 0.749i$
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: \((2,\ 825,\ (\ :1/2),\ -0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46495 + 3.24851i\)
\(L(\frac12)\) \(\approx\) \(1.46495 + 3.24851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.292i)T \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-1.70 - 1.70i)T + 2iT^{2} \)
7 \( 1 + (3.41 - 3.41i)T - 7iT^{2} \)
13 \( 1 + (0.585 + 0.585i)T + 13iT^{2} \)
17 \( 1 + (2 + 2i)T + 17iT^{2} \)
19 \( 1 - 4.82iT - 19T^{2} \)
23 \( 1 + (-4.82 + 4.82i)T - 23iT^{2} \)
29 \( 1 - 3.17T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-5.65 + 5.65i)T - 37iT^{2} \)
41 \( 1 - 0.828iT - 41T^{2} \)
43 \( 1 + (7.41 + 7.41i)T + 43iT^{2} \)
47 \( 1 + (-0.828 - 0.828i)T + 47iT^{2} \)
53 \( 1 + (8.48 - 8.48i)T - 53iT^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-5.41 + 5.41i)T - 67iT^{2} \)
71 \( 1 - 1.65iT - 71T^{2} \)
73 \( 1 + (7.41 + 7.41i)T + 73iT^{2} \)
79 \( 1 + 0.828iT - 79T^{2} \)
83 \( 1 + (-4.24 + 4.24i)T - 83iT^{2} \)
89 \( 1 + 7.31T + 89T^{2} \)
97 \( 1 + (9.65 - 9.65i)T - 97iT^{2} \)
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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

−10.26869131349604151187868846536, −9.349631548739367769439685902509, −8.643225913383121025374583650701, −7.81697069055026479364788467699, −6.79903078188632046789956573932, −6.33829448713406888952126886621, −5.32051823551166086005675653144, −4.28032781480575632462535391856, −3.21311261623855890029700422658, −2.51512906925252361408136591403, 1.16832028659793016481114662766, 2.72545962497927793317136682411, 3.31112885931583810721935537207, 4.15104863010959845932273260243, 4.89930764233685310221701021580, 6.41518835879857036875369705868, 7.10527903962260321306404133691, 8.378072226039404896631782526398, 9.649328645596489094213503123352, 9.892777458764268740809893070349

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