Properties

Label 2-825-275.71-c1-0-45
Degree $2$
Conductor $825$
Sign $0.955 - 0.294i$
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  + 2.23·2-s + (0.309 + 0.951i)3-s + 3.00·4-s + (1.80 − 1.31i)5-s + (0.690 + 2.12i)6-s + (−0.309 + 0.224i)7-s + 2.23·8-s + (−0.809 + 0.587i)9-s + (4.04 − 2.93i)10-s + (−1.69 + 2.85i)11-s + (0.927 + 2.85i)12-s + (4.85 − 3.52i)13-s + (−0.690 + 0.502i)14-s + (1.80 + 1.31i)15-s − 0.999·16-s + (0.545 + 1.67i)17-s + ⋯
L(s)  = 1  + 1.58·2-s + (0.178 + 0.549i)3-s + 1.50·4-s + (0.809 − 0.587i)5-s + (0.282 + 0.868i)6-s + (−0.116 + 0.0848i)7-s + 0.790·8-s + (−0.269 + 0.195i)9-s + (1.27 − 0.929i)10-s + (−0.509 + 0.860i)11-s + (0.267 + 0.823i)12-s + (1.34 − 0.978i)13-s + (−0.184 + 0.134i)14-s + (0.467 + 0.339i)15-s − 0.249·16-s + (0.132 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.955 - 0.294i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.955 - 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.23016 + 0.636893i\)
\(L(\frac12)\) \(\approx\) \(4.23016 + 0.636893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 1.31i)T \)
11 \( 1 + (1.69 - 2.85i)T \)
good2 \( 1 - 2.23T + 2T^{2} \)
7 \( 1 + (0.309 - 0.224i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.85 + 3.52i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.545 - 1.67i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 - 4.38T + 19T^{2} \)
23 \( 1 + (1.07 - 3.30i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + 4.09T + 29T^{2} \)
31 \( 1 + (-1.19 - 0.865i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.35 + 4.61i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.92 + 5.93i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.94T + 43T^{2} \)
47 \( 1 + (0.954 + 2.93i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.38 - 1.00i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.35 + 6.06i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.690 - 0.502i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-1.19 + 0.865i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.381 - 1.17i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.52 + 4.70i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.57 - 10.9i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.11 - 15.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.54 + 10.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.37785867480025746552953034472, −9.549086535203701404990535652590, −8.646059142007551422975311255706, −7.54731329179566735120222918326, −6.29338388451645236116348755665, −5.48603350027895258420344376082, −5.08829855249410590591445932343, −3.90278339876144726664942241811, −3.12473936808948237651006938590, −1.81867664188793009421331388960, 1.70192767795854775939820976562, 2.95197217021568788619982616367, 3.55325973405149713124380433055, 4.92544126136336758492385159175, 5.86887578601019669462475775603, 6.39701039959756536287318615919, 7.14463894800781613131640615160, 8.396918117329405777308441742684, 9.348953805867547470985561967157, 10.47634202761858743757941536274

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