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Properties

Label 2-1975-1975.789-c0-0-1
Degree $2$
Conductor $1975$
Sign $0.637 + 0.770i$
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  + (−1.17 − 1.61i)2-s + (−0.922 + 2.83i)4-s + (0.876 + 0.481i)5-s + (3.76 − 1.22i)8-s + (0.809 + 0.587i)9-s + (−0.250 − 1.98i)10-s + (1.56 − 1.13i)11-s + (−0.905 + 1.24i)13-s + (−3.98 − 2.89i)16-s − 1.99i·18-s + (0.115 + 0.356i)19-s + (−2.17 + 2.04i)20-s + (−3.67 − 1.19i)22-s + (−0.566 − 0.779i)23-s + (0.535 + 0.844i)25-s + 3.07·26-s + ⋯
L(s)  = 1  + (−1.17 − 1.61i)2-s + (−0.922 + 2.83i)4-s + (0.876 + 0.481i)5-s + (3.76 − 1.22i)8-s + (0.809 + 0.587i)9-s + (−0.250 − 1.98i)10-s + (1.56 − 1.13i)11-s + (−0.905 + 1.24i)13-s + (−3.98 − 2.89i)16-s − 1.99i·18-s + (0.115 + 0.356i)19-s + (−2.17 + 2.04i)20-s + (−3.67 − 1.19i)22-s + (−0.566 − 0.779i)23-s + (0.535 + 0.844i)25-s + 3.07·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1975\)    =    \(5^{2} \cdot 79\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(0.985653\)
Root analytic conductor: \(0.992800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1975} (789, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1975,\ (\ :0),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7974483908\)
\(L(\frac12)\) \(\approx\) \(0.7974483908\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.876 - 0.481i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (1.17 + 1.61i)T + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-1.56 + 1.13i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.905 - 1.24i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.115 - 0.356i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.566 + 0.779i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.238 - 0.0774i)T + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.804 + 1.10i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (1.03 - 0.749i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-1.72 - 0.559i)T + (0.809 + 0.587i)T^{2} \)
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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

−9.324734628223096714889857854775, −8.948534849343772524227763336813, −8.018644783166241952918666259695, −7.05480796422555604543365886581, −6.46022785860301054782272224556, −4.76881984796835676563787610437, −3.97922155235585243411096226494, −3.04894588534054485351574346424, −2.00571819485051656983544302875, −1.38464075524229162562464694580, 1.03861538797779602265088763792, 1.94159399725154964812926687168, 4.23192425407292250635940643893, 4.93666578382752043213276790849, 5.81346512526030285602270062683, 6.48518546784766978044023344981, 7.16931495229064885172501835400, 7.77240842151778117079697728859, 8.763340404382850906542503794808, 9.456632298291654362595542912659

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