Properties

Label 2-825-11.3-c1-0-24
Degree $2$
Conductor $825$
Sign $0.993 - 0.117i$
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  + (1.08 − 0.786i)2-s + (0.309 + 0.951i)3-s + (−0.0646 + 0.198i)4-s + (1.08 + 0.786i)6-s + (1.16 − 3.59i)7-s + (0.913 + 2.81i)8-s + (−0.809 + 0.587i)9-s + (−0.569 + 3.26i)11-s − 0.209·12-s + (3.99 − 2.90i)13-s + (−1.56 − 4.81i)14-s + (2.86 + 2.07i)16-s + (2.63 + 1.91i)17-s + (−0.413 + 1.27i)18-s + (0.424 + 1.30i)19-s + ⋯
L(s)  = 1  + (0.765 − 0.556i)2-s + (0.178 + 0.549i)3-s + (−0.0323 + 0.0994i)4-s + (0.442 + 0.321i)6-s + (0.441 − 1.35i)7-s + (0.322 + 0.994i)8-s + (−0.269 + 0.195i)9-s + (−0.171 + 0.985i)11-s − 0.0603·12-s + (1.10 − 0.804i)13-s + (−0.418 − 1.28i)14-s + (0.715 + 0.519i)16-s + (0.639 + 0.464i)17-s + (−0.0974 + 0.299i)18-s + (0.0973 + 0.299i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.993 - 0.117i$
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: \((2,\ 825,\ (\ :1/2),\ 0.993 - 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60393 + 0.153706i\)
\(L(\frac12)\) \(\approx\) \(2.60393 + 0.153706i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.569 - 3.26i)T \)
good2 \( 1 + (-1.08 + 0.786i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-1.16 + 3.59i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.99 + 2.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.63 - 1.91i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.424 - 1.30i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.93T + 23T^{2} \)
29 \( 1 + (0.537 - 1.65i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.79 + 3.48i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.45 - 4.46i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.91 - 8.97i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (0.248 + 0.766i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.65 + 6.28i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.75 + 11.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.95 + 5.04i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.77T + 67T^{2} \)
71 \( 1 + (-2.21 - 1.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.89 + 8.91i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.3 - 7.50i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.76 + 2.00i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.85T + 89T^{2} \)
97 \( 1 + (11.0 - 8.00i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.38467203265550332677037629004, −9.759059531766634160451611795691, −8.204806680288343599044257203628, −8.019537485092587764036315939206, −6.77729282536876596452773738815, −5.41012211129276065192720709214, −4.58977003893069335886947095613, −3.84297121973513386068367342136, −3.08233752389297309132599629287, −1.50057362125587614221507245203, 1.24661202329698550578626382662, 2.72307000469454979865398730937, 3.89960073721599235013605879261, 5.20434297850907296110955691327, 5.76159543265061979520356771057, 6.50068433604105391176103053260, 7.45814081579789193161581679878, 8.692805612409477082236137964112, 8.916599169658708211453422596659, 10.22088883221404636290610810429

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