Properties

Label 2-825-11.9-c1-0-10
Degree $2$
Conductor $825$
Sign $-0.569 - 0.821i$
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  + (−0.5 + 1.53i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (0.5 + 1.53i)6-s + (4.23 + 3.07i)7-s + (−1.80 + 1.31i)8-s + (0.309 − 0.951i)9-s + (−1.23 + 3.07i)11-s − 0.618·12-s + (1 − 3.07i)13-s + (−6.85 + 4.97i)14-s + (−1.50 − 4.61i)16-s + (0.618 + 1.90i)17-s + (1.30 + 0.951i)18-s + (−4.04 + 2.93i)19-s + ⋯
L(s)  = 1  + (−0.353 + 1.08i)2-s + (0.467 − 0.339i)3-s + (−0.250 − 0.181i)4-s + (0.204 + 0.628i)6-s + (1.60 + 1.16i)7-s + (−0.639 + 0.464i)8-s + (0.103 − 0.317i)9-s + (−0.372 + 0.927i)11-s − 0.178·12-s + (0.277 − 0.853i)13-s + (−1.83 + 1.33i)14-s + (−0.375 − 1.15i)16-s + (0.149 + 0.461i)17-s + (0.308 + 0.224i)18-s + (−0.928 + 0.674i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.569 - 0.821i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.810624 + 1.54861i\)
\(L(\frac12)\) \(\approx\) \(0.810624 + 1.54861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 \)
11 \( 1 + (1.23 - 3.07i)T \)
good2 \( 1 + (0.5 - 1.53i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-4.23 - 3.07i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1 + 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.618 - 1.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.04 - 2.93i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 + (0.690 + 0.502i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.16 + 6.65i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.19 + 0.865i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.28 - 6.74i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.09T + 43T^{2} \)
47 \( 1 + (-4.92 + 3.57i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.66 - 5.11i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.42 + 3.94i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.16 - 6.65i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + (2.47 + 7.60i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.66 - 2.66i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.954 + 2.93i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.38 - 7.33i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.14T + 89T^{2} \)
97 \( 1 + (-0.781 + 2.40i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.43378598955463539001273096139, −9.191457511941387872268516617011, −8.507621966372289561825654072964, −7.944615526712871724769293889899, −7.42797023745197001597759160695, −6.17843350486515545370776951254, −5.49778363760124556416854223819, −4.52546932825995211053619351422, −2.77122121547077202295512794960, −1.81688313921970856044095769357, 0.971518130111110476105067919486, 2.06086446188747426896356905546, 3.27270605801694435473804817222, 4.25417018101278678568661365300, 5.11905164334217087234796688373, 6.64127293570398373611064455237, 7.54833011231271547897372926328, 8.650573329216598676843971127247, 8.992195588473421783268293106181, 10.31836919468800715557575831730

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