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Properties

Label 2-825-15.2-c1-0-35
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  + (1.70 − 1.70i)2-s + (0.292 + 1.70i)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 + (6.53 − 1.12i)12-s + (0.585 − 0.585i)13-s + 11.6·14-s − 2.99·16-s + (−2 + 2i)17-s + (−3.12 + 6.53i)18-s − 4.82i·19-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + (0.169 + 0.985i)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 + (1.88 − 0.323i)12-s + (0.162 − 0.162i)13-s + 3.11·14-s − 0.749·16-s + (−0.485 + 0.485i)17-s + (−0.735 + 1.54i)18-s − 1.10i·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: $\chi_{825} (782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.30940 - 0.642739i\)
\(L(\frac12)\) \(\approx\) \(3.30940 - 0.642739i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 1.70i)T - 2iT^{2} \)
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

−10.78647552916999899865282762925, −9.366312677729919294332197546406, −8.943464514065709418224601283543, −7.80507972348164153357928630449, −6.07182482215816929604391781891, −5.11332849751812191610437964309, −4.86433223740248432085658980452, −3.73810728991915115361972205034, −2.69835982543350375541197656656, −1.84428221792995801580173526048, 1.34059754197963433943581209286, 3.05590735776615951004997957920, 4.24775454092205441129578900815, 4.94081429454780466409407770963, 6.06330803803288367995174178024, 6.77504781541599753393350222496, 7.60795764311096886650837953146, 7.976791217153571513852826816793, 8.904907895215222960378682812529, 10.57580944804954225254999497771

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