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Properties

Label 2-3225-129.92-c0-0-0
Degree $2$
Conductor $3225$
Sign $0.216 - 0.976i$
Analytic cond. $1.60948$
Root an. cond. $1.26865$
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  + (−0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.5 + 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)36-s + (0.5 − 0.866i)37-s − 0.999·39-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.5 + 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)36-s + (0.5 − 0.866i)37-s − 0.999·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.216 - 0.976i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (2801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.216 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.463357699\)
\(L(\frac12)\) \(\approx\) \(1.463357699\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
43 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.879245694950213162958463198032, −8.575494196479600590920330931554, −7.25574859047292966512931792571, −6.77492386386103877983141237002, −5.77351158407256581649650935286, −5.38993879012436745614302960187, −4.41790769047916694597336419642, −3.45058250626535864084691293576, −2.57300080859773070905560532182, −1.49357519845321597025467258940, 1.02593431197738091272503440851, 1.83730551745539479161619287248, 2.91975535002468095560665916862, 3.88309802312457573517559292817, 5.08548430920412413234928432688, 5.86726439754902342590044872300, 6.41134778477984270153572810692, 7.26123255470689522668419821031, 7.926278523403895833945735734969, 8.079147809171169484941832813308

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