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Properties

Label 2-950-5.4-c3-0-79
Degree $2$
Conductor $950$
Sign $-0.894 - 0.447i$
Analytic cond. $56.0518$
Root an. cond. $7.48677$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 8.77i·3-s − 4·4-s + 17.5·6-s − 26.0i·7-s − 8i·8-s − 49.9·9-s − 4.22·11-s + 35.0i·12-s − 64.0i·13-s + 52.1·14-s + 16·16-s − 48.5i·17-s − 99.8i·18-s − 19·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.68i·3-s − 0.5·4-s + 1.19·6-s − 1.40i·7-s − 0.353i·8-s − 1.84·9-s − 0.115·11-s + 0.844i·12-s − 1.36i·13-s + 0.996·14-s + 0.250·16-s − 0.692i·17-s − 1.30i·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(56.0518\)
Root analytic conductor: \(7.48677\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.258273509\)
\(L(\frac12)\) \(\approx\) \(1.258273509\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 \)
19 \( 1 + 19T \)
good3 \( 1 + 8.77iT - 27T^{2} \)
7 \( 1 + 26.0iT - 343T^{2} \)
11 \( 1 + 4.22T + 1.33e3T^{2} \)
13 \( 1 + 64.0iT - 2.19e3T^{2} \)
17 \( 1 + 48.5iT - 4.91e3T^{2} \)
23 \( 1 + 92.0iT - 1.21e4T^{2} \)
29 \( 1 - 88.2T + 2.43e4T^{2} \)
31 \( 1 + 81.9T + 2.97e4T^{2} \)
37 \( 1 + 23.6iT - 5.06e4T^{2} \)
41 \( 1 - 17.7T + 6.89e4T^{2} \)
43 \( 1 + 368. iT - 7.95e4T^{2} \)
47 \( 1 + 497. iT - 1.03e5T^{2} \)
53 \( 1 - 536. iT - 1.48e5T^{2} \)
59 \( 1 - 36.6T + 2.05e5T^{2} \)
61 \( 1 - 630.T + 2.26e5T^{2} \)
67 \( 1 - 282. iT - 3.00e5T^{2} \)
71 \( 1 - 595.T + 3.57e5T^{2} \)
73 \( 1 - 597. iT - 3.89e5T^{2} \)
79 \( 1 + 427.T + 4.93e5T^{2} \)
83 \( 1 + 493. iT - 5.71e5T^{2} \)
89 \( 1 - 921.T + 7.04e5T^{2} \)
97 \( 1 - 1.08e3iT - 9.12e5T^{2} \)
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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

−8.703531724635522232119705224143, −8.048123741432045693182626291365, −7.28327853077012124896301170187, −6.94934145795351905011779027930, −5.98692387887418298169259154618, −5.05114150905957198977645903870, −3.71574651256779815468128015624, −2.46182573902933849191810101278, −0.944322427369904137023991526210, −0.38985618159590440381217835783, 1.88228449043159278242082293952, 2.94594442960384008166827057752, 3.89140227424303759456683148788, 4.70680395687328006947044370453, 5.47902944696050874822341675209, 6.38514889236278800318774446612, 8.124854913160828620071956032143, 8.913012127932384475182393732134, 9.406090674989947126096576785238, 9.989865424483033355803084782931

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