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Properties

Label 2-825-165.38-c1-0-34
Degree $2$
Conductor $825$
Sign $-0.0669 - 0.997i$
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  + (−1.15 + 2.27i)2-s + (1.70 + 0.320i)3-s + (−2.64 − 3.63i)4-s + (−2.69 + 3.49i)6-s + (0.237 − 1.50i)7-s + (6.27 − 0.993i)8-s + (2.79 + 1.09i)9-s + (−0.991 + 3.16i)11-s + (−3.32 − 7.03i)12-s + (1.69 − 3.32i)13-s + (3.13 + 2.27i)14-s + (−2.22 + 6.84i)16-s + (4.74 − 2.42i)17-s + (−5.70 + 5.08i)18-s + (−0.443 + 0.610i)19-s + ⋯
L(s)  = 1  + (−0.817 + 1.60i)2-s + (0.982 + 0.185i)3-s + (−1.32 − 1.81i)4-s + (−1.10 + 1.42i)6-s + (0.0898 − 0.567i)7-s + (2.21 − 0.351i)8-s + (0.931 + 0.363i)9-s + (−0.298 + 0.954i)11-s + (−0.961 − 2.02i)12-s + (0.469 − 0.922i)13-s + (0.837 + 0.608i)14-s + (−0.555 + 1.71i)16-s + (1.15 − 0.586i)17-s + (−1.34 + 1.19i)18-s + (−0.101 + 0.140i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.0669 - 0.997i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.0669 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.972667 + 1.04010i\)
\(L(\frac12)\) \(\approx\) \(0.972667 + 1.04010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.320i)T \)
5 \( 1 \)
11 \( 1 + (0.991 - 3.16i)T \)
good2 \( 1 + (1.15 - 2.27i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (-0.237 + 1.50i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-1.69 + 3.32i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-4.74 + 2.42i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.443 - 0.610i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-4.86 + 4.86i)T - 23iT^{2} \)
29 \( 1 + (-3.88 + 2.82i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.607 + 1.87i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.53 - 9.70i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (4.14 - 5.69i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (7.47 + 7.47i)T + 43iT^{2} \)
47 \( 1 + (-1.66 - 10.4i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-7.14 - 3.63i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (8.99 - 6.53i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.364 - 1.12i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-11.0 + 11.0i)T - 67iT^{2} \)
71 \( 1 + (0.382 + 0.124i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.52 - 0.399i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-2.47 + 0.804i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.72 + 5.35i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 1.88T + 89T^{2} \)
97 \( 1 + (1.33 + 0.678i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.12972384387118335611613054839, −9.403602452669110705499456077554, −8.461232643712878929986235925630, −7.904173691186492892018720074429, −7.27869595413665625734377777320, −6.47854992522153751382657744507, −5.20995735483362887224460104730, −4.47519481670564406112192347962, −2.99482714358968927263786140978, −1.10945254836536785881935432356, 1.19164987606182542911807859316, 2.18948512139555992778786163122, 3.28026184668732491356413198116, 3.79798406378689339168289052681, 5.34109138173586237046455864361, 6.90224526136507366993301708130, 8.020013503546648474441985448208, 8.672565633211254855026314323632, 9.101585761492886134513610683369, 9.987846605801190906258971512460

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