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Properties

Label 2-2925-325.233-c0-0-2
Degree $2$
Conductor $2925$
Sign $0.770 - 0.637i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
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  + (−0.905 + 1.77i)2-s + (−1.74 − 2.40i)4-s + (0.233 − 0.972i)5-s + (3.89 − 0.616i)8-s + (1.51 + 1.29i)10-s + (1.62 − 0.526i)11-s + (−0.891 + 0.453i)13-s + (−1.50 + 4.63i)16-s + (−2.74 + 1.13i)20-s + (−0.531 + 3.35i)22-s + (−0.891 − 0.453i)25-s − 1.99i·26-s + (−4.09 − 4.09i)32-s + (0.309 − 3.92i)40-s + (1.23 + 0.401i)41-s + ⋯
L(s)  = 1  + (−0.905 + 1.77i)2-s + (−1.74 − 2.40i)4-s + (0.233 − 0.972i)5-s + (3.89 − 0.616i)8-s + (1.51 + 1.29i)10-s + (1.62 − 0.526i)11-s + (−0.891 + 0.453i)13-s + (−1.50 + 4.63i)16-s + (−2.74 + 1.13i)20-s + (−0.531 + 3.35i)22-s + (−0.891 − 0.453i)25-s − 1.99i·26-s + (−4.09 − 4.09i)32-s + (0.309 − 3.92i)40-s + (1.23 + 0.401i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7455204301\)
\(L(\frac12)\) \(\approx\) \(0.7455204301\)
\(L(1)\) 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 \( 1 + (-0.233 + 0.972i)T \)
13 \( 1 + (0.891 - 0.453i)T \)
good2 \( 1 + (0.905 - 1.77i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-1.62 + 0.526i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-1.23 - 0.401i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
47 \( 1 + (-1.28 - 0.203i)T + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.600 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.614 + 0.845i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.154 - 0.0245i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 + (-0.236 - 0.727i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.951 + 0.309i)T^{2} \)
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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

−8.999241772378503769208311514839, −8.274678834066383640209042148385, −7.56409132828788234410069452479, −6.76607143359227086904455624768, −6.15561829168844760584641584794, −5.47557370496661680020603072915, −4.61704298579438135392472382216, −4.03886326948059344650544708002, −1.79230197617345158697908320532, −0.791670655151930454368025392761, 1.17176060204046382492103617322, 2.25473980947417168051867837590, 2.85116719649836405488487301954, 3.89307004222280925428439199635, 4.36334390619687527696177589477, 5.72485195981007329422257471726, 7.15976755259142027292909688159, 7.32685852396441919754896542926, 8.444360050824015975024665162974, 9.268979034671269726016769977821

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