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Properties

Label 2-60e2-5.4-c1-0-16
Degree $2$
Conductor $3600$
Sign $0.447 - 0.894i$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
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  + 4i·7-s − 2i·13-s + 8·19-s + 4·31-s − 10i·37-s + 8i·43-s − 9·49-s + 14·61-s + 16i·67-s + 10i·73-s − 4·79-s + 8·91-s + 14i·97-s + 20i·103-s − 2·109-s + ⋯
L(s)  = 1  + 1.51i·7-s − 0.554i·13-s + 1.83·19-s + 0.718·31-s − 1.64i·37-s + 1.21i·43-s − 1.28·49-s + 1.79·61-s + 1.95i·67-s + 1.17i·73-s − 0.450·79-s + 0.838·91-s + 1.42i·97-s + 1.97i·103-s − 0.191·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.881224702\)
\(L(\frac12)\) \(\approx\) \(1.881224702\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 16iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.679854463838395859295167662628, −8.016459470359899075972712848442, −7.28101304657493281489624231555, −6.34501718621439964387021312327, −5.46019200520156091419249562695, −5.27436891853782060412211726730, −3.99678718250803551140047687598, −2.97711038721379082663734096602, −2.38245825924528526673071130489, −1.06556667006186634666569473452, 0.67429971704453545005553088813, 1.61928431382313917897956963915, 3.02275634312633459334619558189, 3.74246408996166194370495845381, 4.57373705258790200882689187094, 5.27390804844714373075996193123, 6.36598383165525711690532783933, 7.03241617073278871019508592207, 7.57586021832068742494217947493, 8.302179430291873342597403143302

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