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Properties

Label 2-3825-425.373-c0-0-1
Degree $2$
Conductor $3825$
Sign $-0.612 + 0.790i$
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  + (0.951 + 0.309i)4-s + (−0.707 − 0.707i)5-s + (−1.04 − 1.44i)11-s + (−0.221 − 1.39i)13-s + (0.809 + 0.587i)16-s + (−0.453 + 0.891i)17-s + (−1.80 + 0.587i)19-s + (−0.453 − 0.891i)20-s + (0.253 − 1.59i)23-s + 1.00i·25-s + (0.0966 − 0.297i)29-s + (−1.16 + 1.59i)41-s + (−1.26 − 1.26i)43-s + (−0.550 − 1.69i)44-s − i·49-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)4-s + (−0.707 − 0.707i)5-s + (−1.04 − 1.44i)11-s + (−0.221 − 1.39i)13-s + (0.809 + 0.587i)16-s + (−0.453 + 0.891i)17-s + (−1.80 + 0.587i)19-s + (−0.453 − 0.891i)20-s + (0.253 − 1.59i)23-s + 1.00i·25-s + (0.0966 − 0.297i)29-s + (−1.16 + 1.59i)41-s + (−1.26 − 1.26i)43-s + (−0.550 − 1.69i)44-s − i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3825\)    =    \(3^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(1.90892\)
Root analytic conductor: \(1.38163\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3825} (1648, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3825,\ (\ :0),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8084211876\)
\(L(\frac12)\) \(\approx\) \(0.8084211876\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.453 - 0.891i)T \)
good2 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (1.04 + 1.44i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.253 + 1.59i)T + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (-0.0966 + 0.297i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (1.16 - 1.59i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \)
71 \( 1 + (-0.863 - 0.280i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.587 - 0.809i)T^{2} \)
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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

−8.354207682314166232514977768457, −8.057891787192973826466735698641, −6.91349162470613645071355142449, −6.19092523379842913755185532894, −5.52206079184187468514579320704, −4.60028498892241512690712155636, −3.60243647180263063360905885859, −2.94988492985729495277573857517, −1.94795149676715360014960358015, −0.39917591134749582814977602247, 1.91550276702309706992541623953, 2.37497539592844245877468238814, 3.38952305492885310031197366481, 4.49703252056784290227102260199, 5.02039669588280178022171188499, 6.25247751793851784716018525973, 6.99143576761472693972477350464, 7.16548546388532752361065812668, 7.939796387725717872548636807485, 8.965513947505217885190907031609

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