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Properties

Label 2-1200-60.59-c1-0-8
Degree $2$
Conductor $1200$
Sign $-0.316 - 0.948i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
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  + (−1.22 + 1.22i)3-s + 2.44·7-s − 2.99i·9-s − 4.89·11-s + 2i·13-s + 6·17-s + 4.89i·19-s + (−2.99 + 2.99i)21-s − 2.44i·23-s + (3.67 + 3.67i)27-s + 9.79i·31-s + (5.99 − 5.99i)33-s + 2i·37-s + (−2.44 − 2.44i)39-s − 6i·41-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + 0.925·7-s − 0.999i·9-s − 1.47·11-s + 0.554i·13-s + 1.45·17-s + 1.12i·19-s + (−0.654 + 0.654i)21-s − 0.510i·23-s + (0.707 + 0.707i)27-s + 1.75i·31-s + (1.04 − 1.04i)33-s + 0.328i·37-s + (−0.392 − 0.392i)39-s − 0.937i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.080217738\)
\(L(\frac12)\) \(\approx\) \(1.080217738\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 - 2.44T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + 2.44iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 9.79iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 2.44T + 43T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 7.34T + 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 4.89iT - 79T^{2} \)
83 \( 1 - 7.34iT - 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.23792399073936791352990384661, −9.334402142277235887891941927201, −8.254352275886005676073591755565, −7.70099128026319700180002899415, −6.53826806688443484105150056407, −5.46293762174267451844887393495, −5.08141218847295693075234329865, −4.05820026651018439650738125514, −2.93856843572055286182055090923, −1.35688950941579948201508818312, 0.54377063936868461743456342907, 1.93652270288700147074454678191, 3.02754418150256193495200735190, 4.65964101672166786085914853699, 5.34134021971849280562829356947, 5.94465182253247385241349778986, 7.25001126726232362392264026358, 7.79831438691005220718335707384, 8.294967596583936632103026081637, 9.684745341005508719579288884442

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