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Properties

Label 2-2925-325.298-c0-0-3
Degree $2$
Conductor $2925$
Sign $0.992 - 0.125i$
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  + (1.92 + 0.304i)2-s + (2.64 + 0.859i)4-s + (−0.649 − 0.760i)5-s + (3.08 + 1.57i)8-s + (−1.01 − 1.65i)10-s + (−0.0922 − 0.126i)11-s + (−0.156 − 0.987i)13-s + (3.20 + 2.32i)16-s + (−1.06 − 2.57i)20-s + (−0.138 − 0.271i)22-s + (−0.156 + 0.987i)25-s − 1.94i·26-s + (2.99 + 2.99i)32-s + (−0.809 − 3.36i)40-s + (−1.00 + 1.37i)41-s + ⋯
L(s)  = 1  + (1.92 + 0.304i)2-s + (2.64 + 0.859i)4-s + (−0.649 − 0.760i)5-s + (3.08 + 1.57i)8-s + (−1.01 − 1.65i)10-s + (−0.0922 − 0.126i)11-s + (−0.156 − 0.987i)13-s + (3.20 + 2.32i)16-s + (−1.06 − 2.57i)20-s + (−0.138 − 0.271i)22-s + (−0.156 + 0.987i)25-s − 1.94i·26-s + (2.99 + 2.99i)32-s + (−0.809 − 3.36i)40-s + (−1.00 + 1.37i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.699262657\)
\(L(\frac12)\) \(\approx\) \(3.699262657\)
\(L(1)\) 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 \( 1 + (0.649 + 0.760i)T \)
13 \( 1 + (0.156 + 0.987i)T \)
good2 \( 1 + (-1.92 - 0.304i)T + (0.951 + 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.0922 + 0.126i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (1.00 - 1.37i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
47 \( 1 + (1.51 - 0.774i)T + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (1.23 + 0.893i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (1.89 + 0.616i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.416 - 0.211i)T + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-1.49 + 1.08i)T + (0.309 - 0.951i)T^{2} \)
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

−8.593177431068747762846139085420, −7.82289782207495030240247397767, −7.43219969975256470041909858019, −6.33178673840538279710459379803, −5.78505863202268077465146510474, −4.80267594742388566495220687363, −4.58067070367058596427726237318, −3.42809822405306462350921356127, −2.94389952604154027207391387652, −1.55631836423140476632283919818, 1.78806721620637459676360570761, 2.65252083549243619513702456760, 3.49970872888261452367921825424, 4.15842071747390554684499980602, 4.83319618748365289819059567679, 5.77446026257248302450187348052, 6.50473652021456805916772589684, 7.14025390320300323835625997377, 7.66094136680378497075648456872, 8.929851869462109058544003722430

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