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Properties

Label 2-1200-15.2-c1-0-26
Degree $2$
Conductor $1200$
Sign $-0.662 + 0.749i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.292 + 1.70i)3-s + (−2 − 2i)7-s + (−2.82 − i)9-s + 5.65i·11-s + (2.82 − 2.82i)17-s + 4i·19-s + (4 − 2.82i)21-s + (−4.24 − 4.24i)23-s + (2.53 − 4.53i)27-s − 5.65·29-s − 8·31-s + (−9.65 − 1.65i)33-s + (−8 − 8i)37-s − 5.65i·41-s + (−2 + 2i)43-s + ⋯
L(s)  = 1  + (−0.169 + 0.985i)3-s + (−0.755 − 0.755i)7-s + (−0.942 − 0.333i)9-s + 1.70i·11-s + (0.685 − 0.685i)17-s + 0.917i·19-s + (0.872 − 0.617i)21-s + (−0.884 − 0.884i)23-s + (0.487 − 0.872i)27-s − 1.05·29-s − 1.43·31-s + (−1.68 − 0.288i)33-s + (−1.31 − 1.31i)37-s − 0.883i·41-s + (−0.304 + 0.304i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 - 1.70i)T \)
5 \( 1 \)
good7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (4.24 + 4.24i)T + 23iT^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (8 + 8i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \)
53 \( 1 + (-5.65 - 5.65i)T + 53iT^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (6 + 6i)T + 67iT^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (8 - 8i)T - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + (9.89 + 9.89i)T + 83iT^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (-8 - 8i)T + 97iT^{2} \)
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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

−9.561461758858264014432520071591, −8.979428040061995763163173640277, −7.62934124188293452706202665145, −7.08865658340810206136314088578, −5.95040283240010447057077064634, −5.10517222830260620554518905651, −4.08462660640692026455977735693, −3.54269177073260169902359654027, −2.08122662945271656368633171479, 0, 1.55996913568475495543508443069, 2.88691164004279061749837348432, 3.59892778277536946967825673883, 5.47889022127029328100287014697, 5.79472229497362126952728517986, 6.64770209205872827267057240311, 7.57115418735111793086113010730, 8.483409082891017258129605252004, 8.962281830009273131701615833614

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