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Properties

Label 2-60e2-20.3-c1-0-37
Degree $2$
Conductor $3600$
Sign $0.0299 + 0.999i$
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  + (1.41 + 1.41i)7-s − 3.46i·11-s + (−2.44 − 2.44i)13-s + (4.89 − 4.89i)17-s + 3.46·19-s − 3.46i·31-s + (−7.34 + 7.34i)37-s − 6·41-s + (−5.65 + 5.65i)43-s + (−8.48 − 8.48i)47-s − 2.99i·49-s + (4.89 + 4.89i)53-s + 10.3·59-s − 10·61-s + (2.82 + 2.82i)67-s + ⋯
L(s)  = 1  + (0.534 + 0.534i)7-s − 1.04i·11-s + (−0.679 − 0.679i)13-s + (1.18 − 1.18i)17-s + 0.794·19-s − 0.622i·31-s + (−1.20 + 1.20i)37-s − 0.937·41-s + (−0.862 + 0.862i)43-s + (−1.23 − 1.23i)47-s − 0.428i·49-s + (0.672 + 0.672i)53-s + 1.35·59-s − 1.28·61-s + (0.345 + 0.345i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0299 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0299 + 0.999i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ 0.0299 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594708324\)
\(L(\frac12)\) \(\approx\) \(1.594708324\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 13iT^{2} \)
17 \( 1 + (-4.89 + 4.89i)T - 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (7.34 - 7.34i)T - 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + (8.48 + 8.48i)T + 47iT^{2} \)
53 \( 1 + (-4.89 - 4.89i)T + 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (4.89 + 4.89i)T + 73iT^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 + 18iT - 89T^{2} \)
97 \( 1 + (4.89 - 4.89i)T - 97iT^{2} \)
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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

−8.321476042373532828399051904635, −7.73053322900470874262635114170, −6.99258241963546803801634436032, −5.98310585672839360121494513114, −5.23818137689236007183115967100, −4.88421661852499026561761990506, −3.34914448718187014150590627484, −3.00683598801546426447724332469, −1.70756799878438561202021729971, −0.48462373685464224520059076901, 1.31577134334223090531173835472, 2.06027238061809185528145765373, 3.36565927867072754931802434171, 4.12000064011638228141190086864, 4.97385474255258638343311519999, 5.56008349587524523739406262985, 6.76209400391355572825068271877, 7.21430463556654580734649835515, 7.938283082554228289619537022484, 8.629457106838201335344089596860

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