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Properties

Label 2-1975-1975.1579-c0-0-1
Degree $2$
Conductor $1975$
Sign $-0.992 + 0.125i$
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  + (−1.72 − 0.559i)2-s + (1.84 + 1.33i)4-s + (−0.929 − 0.368i)5-s + (−1.35 − 1.86i)8-s + (−0.309 − 0.951i)9-s + (1.39 + 1.15i)10-s + (0.115 − 0.356i)11-s + (−0.238 + 0.0774i)13-s + (0.587 + 1.80i)16-s + 1.80i·18-s + (1.56 − 1.13i)19-s + (−1.21 − 1.92i)20-s + (−0.398 + 0.548i)22-s + (−0.700 − 0.227i)23-s + (0.728 + 0.684i)25-s + 0.453·26-s + ⋯
L(s)  = 1  + (−1.72 − 0.559i)2-s + (1.84 + 1.33i)4-s + (−0.929 − 0.368i)5-s + (−1.35 − 1.86i)8-s + (−0.309 − 0.951i)9-s + (1.39 + 1.15i)10-s + (0.115 − 0.356i)11-s + (−0.238 + 0.0774i)13-s + (0.587 + 1.80i)16-s + 1.80i·18-s + (1.56 − 1.13i)19-s + (−1.21 − 1.92i)20-s + (−0.398 + 0.548i)22-s + (−0.700 − 0.227i)23-s + (0.728 + 0.684i)25-s + 0.453·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1975\)    =    \(5^{2} \cdot 79\)
Sign: $-0.992 + 0.125i$
Analytic conductor: \(0.985653\)
Root analytic conductor: \(0.992800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1975} (1579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1975,\ (\ :0),\ -0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2355860650\)
\(L(\frac12)\) \(\approx\) \(0.2355860650\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.929 + 0.368i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.72 + 0.559i)T + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.115 + 0.356i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.238 - 0.0774i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-1.56 + 1.13i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.700 + 0.227i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.905 + 1.24i)T + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.60 + 0.521i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.613 + 1.88i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.17 + 1.61i)T + (-0.309 - 0.951i)T^{2} \)
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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

−8.971673099507485395965726287373, −8.533574950367615031442858361998, −7.51428753101717911259466310781, −7.19107835549719468084392509036, −6.12281318034238025108501663764, −4.83225626019950319932515653737, −3.50564738072406806962525072840, −2.97439497140828340748564405725, −1.46034644070399802226184940259, −0.32030292074128876572398134549, 1.49105257535351380591426004093, 2.63685173282530862176739603439, 3.88407077253096922945085524219, 5.23917122280085242249899331287, 6.03138030531642952723982237073, 7.09640396296575271370919327661, 7.66412159517288221737944097035, 7.966991951574111728982140600946, 8.825813036942829710919544066625, 9.727108493921193261032908971824

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