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Properties

Label 2-3825-17.3-c0-0-1
Degree $2$
Conductor $3825$
Sign $-0.0101 + 0.999i$
Analytic cond. $1.90892$
Root an. cond. $1.38163$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.81 − 0.750i)2-s + (2.01 − 2.01i)4-s + (1.38 − 3.34i)8-s − 4.26i·16-s + (0.980 + 0.195i)17-s + (−0.707 + 0.292i)19-s + (−1.38 + 0.275i)23-s + (−0.324 + 1.63i)31-s + (−1.81 − 4.37i)32-s + (1.92 − 0.382i)34-s + (−1.06 + 1.06i)38-s + (−2.30 + 1.54i)46-s + (−0.785 − 0.785i)47-s + (0.382 + 0.923i)49-s + (0.360 − 0.149i)53-s + ⋯
L(s)  = 1  + (1.81 − 0.750i)2-s + (2.01 − 2.01i)4-s + (1.38 − 3.34i)8-s − 4.26i·16-s + (0.980 + 0.195i)17-s + (−0.707 + 0.292i)19-s + (−1.38 + 0.275i)23-s + (−0.324 + 1.63i)31-s + (−1.81 − 4.37i)32-s + (1.92 − 0.382i)34-s + (−1.06 + 1.06i)38-s + (−2.30 + 1.54i)46-s + (−0.785 − 0.785i)47-s + (0.382 + 0.923i)49-s + (0.360 − 0.149i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3825\)    =    \(3^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.0101 + 0.999i$
Analytic conductor: \(1.90892\)
Root analytic conductor: \(1.38163\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3825} (2026, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3825,\ (\ :0),\ -0.0101 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.878527165\)
\(L(\frac12)\) \(\approx\) \(3.878527165\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 + (-0.980 - 0.195i)T \)
good2 \( 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.923 + 0.382i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (1.38 - 0.275i)T + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.324 - 1.63i)T + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.382 - 0.923i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
53 \( 1 + (-0.360 + 0.149i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.149 + 0.360i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.382 - 0.923i)T^{2} \)
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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

−8.407963933500221585911628836934, −7.44660650353347441285321951084, −6.68542353342539568661697716936, −5.94076005907018419550714866034, −5.40911142678229820156739164798, −4.57937316324487643025428502059, −3.81321154147646523932739508184, −3.21662155738462209255062020347, −2.20489175276601220627153245670, −1.37775794673591043216784696420, 1.99141159140605201018449142941, 2.78801082466387588189093958376, 3.79688421176242545531668286824, 4.24769197545962628508901739524, 5.19630340622550365728884786438, 5.79447798846833498408851343255, 6.42826606835653732377961612056, 7.14246037614310989904548802725, 7.937076387287994141446677582977, 8.318244432622161829086752134380

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