Properties

Label 2-1950-5.4-c1-0-12
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 + 2i·7-s i·8-s − 9-s + 4·11-s i·12-s + i·13-s − 2·14-s + 16-s + 8i·17-s i·18-s + 6·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s − 0.288i·12-s + 0.277i·13-s − 0.534·14-s + 0.250·16-s + 1.94i·17-s − 0.235i·18-s + 1.37·19-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

\(L(1)\) \(\approx\) \(1.670077939\)
\(L(\frac12)\) \(\approx\) \(1.670077939\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−9.266852972621859906261207318532, −8.773660749318981909745276051264, −8.129834527985908072697693270206, −7.06738334483959098178049173674, −6.16734004398887691673910083590, −5.76272696304978056234992772970, −4.60241149650481552731327675004, −3.98457387246634535008300508367, −2.91304486896806243263708883554, −1.40899513089758659861335663546, 0.68983782162457881447637227363, 1.53772617662272086988607589872, 2.93028893992715558285640521905, 3.60063119545053609521505864535, 4.72266480019808520811833735313, 5.50212621464203298033577285997, 6.67978329200774810527219598183, 7.30537283870904236891044208165, 7.978446355489964497448618663604, 9.201104856662972777511994389222

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