Properties

Label 2-1950-65.64-c1-0-26
Degree $2$
Conductor $1950$
Sign $0.124 + 0.992i$
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  − 2-s i·3-s + 4-s + i·6-s + 2·7-s − 8-s − 9-s i·12-s + (3 − 2i)13-s − 2·14-s + 16-s − 2i·17-s + 18-s + i·19-s − 2i·21-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 0.755·7-s − 0.353·8-s − 0.333·9-s − 0.288i·12-s + (0.832 − 0.554i)13-s − 0.534·14-s + 0.250·16-s − 0.485i·17-s + 0.235·18-s + 0.229i·19-s − 0.436i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.124 + 0.992i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.300541978\)
\(L(\frac12)\) \(\approx\) \(1.300541978\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (-3 + 2i)T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 - 5iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 7T + 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.786149615449507024000656424940, −8.234306397243358122292247304650, −7.62037735314479048695969179586, −6.79609357676426913739945188046, −5.96577361513691795225601571884, −5.15318263255546329775953306721, −3.94055344507768358818709446771, −2.76599574179595186235220927712, −1.75940078012088225073514153953, −0.67403979531714114889609173213, 1.21602070870328989511270092443, 2.30369716259958318794779535811, 3.59162767496938085843594843250, 4.36723014899708923145539651733, 5.46041752957976620860636171966, 6.16221352636380788981600596621, 7.21575761657168747962306120766, 7.954416493368725825981979816299, 8.699895026832717351423229167487, 9.289652542310655931348926483791

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