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Properties

Label 2-1975-1975.1184-c0-0-1
Degree $2$
Conductor $1975$
Sign $0.929 + 0.368i$
Analytic cond. $0.985653$
Root an. cond. $0.992800$
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  + (−0.432 + 0.595i)2-s + (0.141 + 0.435i)4-s + (−0.992 − 0.125i)5-s + (−1.02 − 0.331i)8-s + (0.809 − 0.587i)9-s + (0.503 − 0.536i)10-s + (0.101 + 0.0738i)11-s + (−0.992 − 1.36i)13-s + (0.268 − 0.195i)16-s + 0.736i·18-s + (0.263 − 0.809i)19-s + (−0.0858 − 0.449i)20-s + (−0.0879 + 0.0285i)22-s + (−0.147 + 0.202i)23-s + (0.968 + 0.248i)25-s + 1.24·26-s + ⋯
L(s)  = 1  + (−0.432 + 0.595i)2-s + (0.141 + 0.435i)4-s + (−0.992 − 0.125i)5-s + (−1.02 − 0.331i)8-s + (0.809 − 0.587i)9-s + (0.503 − 0.536i)10-s + (0.101 + 0.0738i)11-s + (−0.992 − 1.36i)13-s + (0.268 − 0.195i)16-s + 0.736i·18-s + (0.263 − 0.809i)19-s + (−0.0858 − 0.449i)20-s + (−0.0879 + 0.0285i)22-s + (−0.147 + 0.202i)23-s + (0.968 + 0.248i)25-s + 1.24·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1975\)    =    \(5^{2} \cdot 79\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(0.985653\)
Root analytic conductor: \(0.992800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1975} (1184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1975,\ (\ :0),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6516983945\)
\(L(\frac12)\) \(\approx\) \(0.6516983945\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.992 + 0.125i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.432 - 0.595i)T + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.101 - 0.0738i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.992 + 1.36i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.263 + 0.809i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.147 - 0.202i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (-1.30 - 0.423i)T + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.15 + 1.58i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.866 - 0.629i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.916 - 0.297i)T + (0.809 - 0.587i)T^{2} \)
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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

−9.342604424280732160622805927852, −8.144204619863976393471132402675, −7.86947802037651712806325741087, −7.12181720743329247452539823966, −6.50322341931606277667448435529, −5.34388184483104234081046878948, −4.34432590100482038736458079592, −3.50742661309772351722780700282, −2.62536322533616609167300998132, −0.58077893473890378913025802480, 1.36781627222520576625569855266, 2.34173377189149889268750840142, 3.48769738520977973077119363570, 4.51180191626344791116300200209, 5.17409381040445334225610503327, 6.53527452475407282773589484312, 7.02529613235526924306482750043, 7.949794581656283034756773545862, 8.709565975642335118483230972169, 9.613770581536855709694736408121

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