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Properties

Label 2-2925-325.142-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  + (1.35 + 0.690i)2-s + (0.771 + 1.06i)4-s + (0.852 − 0.522i)5-s + (0.0744 + 0.469i)8-s + (1.51 − 0.119i)10-s + (−0.444 + 0.144i)11-s + (0.453 + 0.891i)13-s + (0.182 − 0.560i)16-s + (1.21 + 0.502i)20-s + (−0.701 − 0.111i)22-s + (0.453 − 0.891i)25-s + 1.52i·26-s + (0.970 − 0.970i)32-s + (0.309 + 0.361i)40-s + (−0.149 − 0.0484i)41-s + ⋯
L(s)  = 1  + (1.35 + 0.690i)2-s + (0.771 + 1.06i)4-s + (0.852 − 0.522i)5-s + (0.0744 + 0.469i)8-s + (1.51 − 0.119i)10-s + (−0.444 + 0.144i)11-s + (0.453 + 0.891i)13-s + (0.182 − 0.560i)16-s + (1.21 + 0.502i)20-s + (−0.701 − 0.111i)22-s + (0.453 − 0.891i)25-s + 1.52i·26-s + (0.970 − 0.970i)32-s + (0.309 + 0.361i)40-s + (−0.149 − 0.0484i)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: $\chi_{2925} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.927130599\)
\(L(\frac12)\) \(\approx\) \(2.927130599\)
\(L(1)\) 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 \( 1 + (-0.852 + 0.522i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
good2 \( 1 + (-1.35 - 0.690i)T + (0.587 + 0.809i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.444 - 0.144i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.149 + 0.0484i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
47 \( 1 + (-0.0245 + 0.154i)T + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.322 - 0.993i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (1.14 + 1.57i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.203 + 1.28i)T + (-0.951 + 0.309i)T^{2} \)
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

−9.009349270676406321525390013314, −8.081913073170449863237903205916, −7.27202858802771147267858023777, −6.31399064751329433291523544198, −6.09349194281320210221011975603, −5.00290365480086665596305554618, −4.68916235497097784175246960714, −3.68703581157711551198907706177, −2.69757517484711679103583067568, −1.54433528620000562130009185966, 1.54918676012708278707948797274, 2.54144927179311847376197545341, 3.17871135772924937563523001533, 3.97880656565468270194487464977, 5.11629445249828008448052932988, 5.53660429215265761044090479044, 6.25751219735400623647223790526, 7.06473251765584033110513574145, 8.130124486343391902407821705798, 8.886634637563682463622552695110

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