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Properties

Label 2-2925-325.38-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  + (−0.163 − 1.03i)2-s + (−0.0875 + 0.0284i)4-s + (−0.0784 − 0.996i)5-s + (−0.430 − 0.845i)8-s + (−1.01 + 0.243i)10-s + (0.763 − 1.05i)11-s + (−0.987 − 0.156i)13-s + (−0.876 + 0.636i)16-s + (0.0352 + 0.0850i)20-s + (−1.20 − 0.616i)22-s + (−0.987 + 0.156i)25-s + 1.04i·26-s + (0.129 + 0.129i)32-s + (−0.809 + 0.495i)40-s + (0.274 + 0.377i)41-s + ⋯
L(s)  = 1  + (−0.163 − 1.03i)2-s + (−0.0875 + 0.0284i)4-s + (−0.0784 − 0.996i)5-s + (−0.430 − 0.845i)8-s + (−1.01 + 0.243i)10-s + (0.763 − 1.05i)11-s + (−0.987 − 0.156i)13-s + (−0.876 + 0.636i)16-s + (0.0352 + 0.0850i)20-s + (−1.20 − 0.616i)22-s + (−0.987 + 0.156i)25-s + 1.04i·26-s + (0.129 + 0.129i)32-s + (−0.809 + 0.495i)40-s + (0.274 + 0.377i)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: $\chi_{2925} (2638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044591212\)
\(L(\frac12)\) \(\approx\) \(1.044591212\)
\(L(1)\) 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 \( 1 + (0.0784 + 0.996i)T \)
13 \( 1 + (0.987 + 0.156i)T \)
good2 \( 1 + (0.163 + 1.03i)T + (-0.951 + 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.763 + 1.05i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.274 - 0.377i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
47 \( 1 + (-0.211 + 0.416i)T + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (-1.44 + 0.469i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.774 + 1.51i)T + (-0.587 + 0.809i)T^{2} \)
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

−8.832251279160140504992428636962, −8.000819954844349493906785249462, −7.06998775636377098694793471734, −6.18247577886179961496367272041, −5.40945617073632327032496596752, −4.41252951903349229034368527704, −3.64458989572618528879443341159, −2.72280894768571647968568423871, −1.67173526000818192826248024326, −0.66841394853604371883014406950, 1.99105566347813869148103934954, 2.75010271312235777763866653827, 3.90733802561006729056750940523, 4.85311686621680606309154941844, 5.73818532131644010283924809394, 6.68141004699284739984308014971, 6.91519422257644437745101068168, 7.65135089027758377228594493826, 8.254300983357303589615058772443, 9.479401933544084752829659108547

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