Properties

Label 2-1200-12.11-c1-0-13
Degree $2$
Conductor $1200$
Sign $-i$
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  + (1.5 + 0.866i)3-s + (1.5 + 2.59i)9-s − 3·11-s − 2·13-s + 5.19i·17-s + 5.19i·19-s + 6·23-s + 5.19i·27-s + 10.3i·29-s − 3.46i·31-s + (−4.5 − 2.59i)33-s + 8·37-s + (−3 − 1.73i)39-s + 5.19i·41-s − 3.46i·43-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.5 + 0.866i)9-s − 0.904·11-s − 0.554·13-s + 1.26i·17-s + 1.19i·19-s + 1.25·23-s + 0.999i·27-s + 1.92i·29-s − 0.622i·31-s + (−0.783 − 0.452i)33-s + 1.31·37-s + (−0.480 − 0.277i)39-s + 0.811i·41-s − 0.528i·43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.921145275\)
\(L(\frac12)\) \(\approx\) \(1.921145275\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 5.19iT - 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 - 5.19iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−9.949071215160877709405285288689, −9.148258902572960841846009259247, −8.255774497202089660090663603470, −7.77666664841938190662447383816, −6.77560219337028756121239169548, −5.54815261604095958672505092947, −4.76447631261619758628405568919, −3.71331411593505908329064024068, −2.85753585158968917578674971710, −1.70698974645208637812633518352, 0.73878341732076462904245968158, 2.48302961494227328366943154877, 2.88253801903961845956274903295, 4.34226208320684842695419885883, 5.18343743253181781866862967579, 6.40687562695412033412906492299, 7.34009431470494632789417767851, 7.71661740425551270066394270788, 8.807714535280942665412841301694, 9.388712944607297374453685683482

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