Properties

Label 2-1200-15.8-c1-0-14
Degree $2$
Conductor $1200$
Sign $0.326 + 0.945i$
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 + (−3.67 + 3.67i)7-s + 2.99i·9-s + (−1.22 − 1.22i)13-s − 7i·19-s + 9·21-s + (3.67 − 3.67i)27-s + 11·31-s + (4.89 − 4.89i)37-s + 2.99i·39-s + (1.22 + 1.22i)43-s − 20i·49-s + (−8.57 + 8.57i)57-s − 61-s + (−11.0 − 11.0i)63-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−1.38 + 1.38i)7-s + 0.999i·9-s + (−0.339 − 0.339i)13-s − 1.60i·19-s + 1.96·21-s + (0.707 − 0.707i)27-s + 1.97·31-s + (0.805 − 0.805i)37-s + 0.480i·39-s + (0.186 + 0.186i)43-s − 2.85i·49-s + (−1.13 + 1.13i)57-s − 0.128·61-s + (−1.38 − 1.38i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.326 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7910190063\)
\(L(\frac12)\) \(\approx\) \(0.7910190063\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.67 - 3.67i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 7iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (-8.57 + 8.57i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (9.79 + 9.79i)T + 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-3.67 + 3.67i)T - 97iT^{2} \)
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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

−9.546572881653600411604133466526, −8.844485665074644456089298205751, −7.88540472422870268425624347373, −6.83731682203099200377387225402, −6.30234720244264038928220051807, −5.55004746568696187073607647491, −4.66070161048579647595081844975, −2.98666061868708644707400269449, −2.35031734485152074039002646127, −0.48230464576594586914570394977, 0.949317840475400380255940887836, 3.03586657001468690597935073141, 3.93053643832338263080004736896, 4.54908460597550949221496140204, 5.86250548661354191679816655373, 6.48149732173987194076299816332, 7.22310816672236021576593933063, 8.293263400848682666183933861284, 9.527914530561104566105047828341, 10.03113149576747701767374655805

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