Properties

Label 2-1950-5.4-c1-0-16
Degree $2$
Conductor $1950$
Sign $0.894 - 0.447i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s i·3-s − 4-s + 6-s i·8-s − 9-s + i·12-s i·13-s + 16-s + 6i·17-s i·18-s − 4i·23-s − 24-s + 26-s + i·27-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 0.277i·13-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s − 0.834i·23-s − 0.204·24-s + 0.196·26-s + 0.192i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.577941544\)
\(L(\frac12)\) \(\approx\) \(1.577941544\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.873004737428168776880637713402, −8.345238335244018553032912949153, −7.76155103767951533411174428588, −6.68956492523292333560510877869, −6.33601106139981271770406232860, −5.38195764772268698999721999460, −4.47907657810209676321139527920, −3.45813428929516679023275408846, −2.24604823741025748567118278636, −0.872231426305974713810506079661, 0.837038914943169770057565526587, 2.34576781878962605888678853675, 3.16236242952289672021194959278, 4.15707191570699309875231884706, 4.89853320263521452178116321586, 5.68444626772792164123977746234, 6.79079837994010600856173324279, 7.67974513181660500879309604141, 8.616922852978376070472245639819, 9.324829414365897403534583934842

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