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Properties

Label 2-825-11.4-c1-0-21
Degree $2$
Conductor $825$
Sign $-0.0694 + 0.997i$
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  + (−0.5 − 0.363i)2-s + (−0.309 + 0.951i)3-s + (−0.5 − 1.53i)4-s + (0.5 − 0.363i)6-s + (−0.236 − 0.726i)7-s + (−0.690 + 2.12i)8-s + (−0.809 − 0.587i)9-s + (3.23 − 0.726i)11-s + 1.61·12-s + (1 + 0.726i)13-s + (−0.145 + 0.449i)14-s + (−1.49 + 1.08i)16-s + (−1.61 + 1.17i)17-s + (0.190 + 0.587i)18-s + (1.54 − 4.75i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.256i)2-s + (−0.178 + 0.549i)3-s + (−0.250 − 0.769i)4-s + (0.204 − 0.148i)6-s + (−0.0892 − 0.274i)7-s + (−0.244 + 0.751i)8-s + (−0.269 − 0.195i)9-s + (0.975 − 0.219i)11-s + 0.467·12-s + (0.277 + 0.201i)13-s + (−0.0389 + 0.120i)14-s + (−0.374 + 0.272i)16-s + (−0.392 + 0.285i)17-s + (0.0450 + 0.138i)18-s + (0.354 − 1.09i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.0694 + 0.997i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.638011 - 0.683991i\)
\(L(\frac12)\) \(\approx\) \(0.638011 - 0.683991i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 0.726i)T \)
good2 \( 1 + (0.5 + 0.363i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.236 + 0.726i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1 - 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.61 - 1.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.54 + 4.75i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.38T + 23T^{2} \)
29 \( 1 + (1.80 + 5.56i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.66 + 4.11i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.30 + 7.10i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.781 + 2.40i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.09T + 43T^{2} \)
47 \( 1 + (-1.57 + 4.84i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.16 - 4.47i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.07 + 6.37i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.66 - 4.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.38T + 67T^{2} \)
71 \( 1 + (-6.47 + 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.16 + 12.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.54 - 4.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.61 + 3.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (9.28 + 6.74i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.983507548551858810137129069526, −9.059251059231923991997904950622, −8.920833296151899040875715942219, −7.41309031391923619252741329920, −6.37580130901964786007775547966, −5.59048646960539845377784768621, −4.56064536858319711260045220679, −3.66534184050959743334984742927, −2.08329293964844608379847082669, −0.58196919480990581816383050627, 1.39499216688172695756885602793, 3.01836473796484838681013901651, 3.96834881154801972310894742040, 5.22893066920209753709822214577, 6.37811955303380402735548935020, 7.07186795255288865495525918424, 7.85062986120323259980960206035, 8.794387246232088330889972686378, 9.257050885506035748575045628320, 10.38014246977664930415984880259

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