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Properties

Label 2-2925-325.142-c0-0-3
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  + (1.15 + 0.589i)2-s + (0.403 + 0.555i)4-s + (0.522 + 0.852i)5-s + (−0.0635 − 0.401i)8-s + (0.101 + 1.29i)10-s + (1.84 − 0.600i)11-s + (−0.453 − 0.891i)13-s + (0.375 − 1.15i)16-s + (−0.262 + 0.634i)20-s + (2.49 + 0.395i)22-s + (−0.453 + 0.891i)25-s − 1.29i·26-s + (0.828 − 0.828i)32-s + (0.309 − 0.263i)40-s + (−1.89 − 0.616i)41-s + ⋯
L(s)  = 1  + (1.15 + 0.589i)2-s + (0.403 + 0.555i)4-s + (0.522 + 0.852i)5-s + (−0.0635 − 0.401i)8-s + (0.101 + 1.29i)10-s + (1.84 − 0.600i)11-s + (−0.453 − 0.891i)13-s + (0.375 − 1.15i)16-s + (−0.262 + 0.634i)20-s + (2.49 + 0.395i)22-s + (−0.453 + 0.891i)25-s − 1.29i·26-s + (0.828 − 0.828i)32-s + (0.309 − 0.263i)40-s + (−1.89 − 0.616i)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

\(L(\frac{1}{2})\) \(\approx\) \(2.583263534\)
\(L(\frac12)\) \(\approx\) \(2.583263534\)
\(L(1)\) 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 \( 1 + (-0.522 - 0.852i)T \)
13 \( 1 + (0.453 + 0.891i)T \)
good2 \( 1 + (-1.15 - 0.589i)T + (0.587 + 0.809i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-1.84 + 0.600i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (1.89 + 0.616i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
47 \( 1 + (0.311 - 1.96i)T + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.526 + 1.62i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.610 - 1.87i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.274 + 0.377i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.237 - 1.50i)T + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (0.570 + 1.75i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.098182239670011356037168553301, −8.048584203055549687599746520894, −7.11892952944682888696857070999, −6.53229070861150711853121305070, −6.05027179676163757768534652139, −5.29523698715655562667147237237, −4.37537665674077374514812875138, −3.48053711388649309027015177947, −2.94003971559015740796570341867, −1.41577573869076551519956011846, 1.58656536598044653139471267701, 2.10013834810098631547088825056, 3.53322899638590916075977739954, 4.11736103215624001838602214325, 4.86920685906737540052790706545, 5.44304895297902550486361414298, 6.52861440546739183141734275462, 6.95249099690014633484912668897, 8.450384667421070928648835679198, 8.789892543413495077282977553189

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