Properties

Label 2-60e2-15.2-c1-0-35
Degree $2$
Conductor $3600$
Sign $-0.998 + 0.0618i$
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  + (−3.23 − 3.23i)7-s − 4.57i·11-s + (4.23 − 4.23i)13-s + (1.74 − 1.74i)17-s + 2.47i·19-s + (−2.82 − 2.82i)23-s − 5.99·29-s − 1.52·31-s + (−2.23 − 2.23i)37-s + 7.07i·41-s + (2.47 − 2.47i)43-s + (1.74 − 1.74i)47-s + 13.9i·49-s + 1.08·59-s + 10.4·61-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)7-s − 1.37i·11-s + (1.17 − 1.17i)13-s + (0.423 − 0.423i)17-s + 0.567i·19-s + (−0.589 − 0.589i)23-s − 1.11·29-s − 0.274·31-s + (−0.367 − 0.367i)37-s + 1.10i·41-s + (0.376 − 0.376i)43-s + (0.254 − 0.254i)47-s + 1.99i·49-s + 0.140·59-s + 1.34·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.998 + 0.0618i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9378753669\)
\(L(\frac12)\) \(\approx\) \(0.9378753669\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 + 3.23i)T + 7iT^{2} \)
11 \( 1 + 4.57iT - 11T^{2} \)
13 \( 1 + (-4.23 + 4.23i)T - 13iT^{2} \)
17 \( 1 + (-1.74 + 1.74i)T - 17iT^{2} \)
19 \( 1 - 2.47iT - 19T^{2} \)
23 \( 1 + (2.82 + 2.82i)T + 23iT^{2} \)
29 \( 1 + 5.99T + 29T^{2} \)
31 \( 1 + 1.52T + 31T^{2} \)
37 \( 1 + (2.23 + 2.23i)T + 37iT^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \)
47 \( 1 + (-1.74 + 1.74i)T - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 - 1.08T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + (1.52 + 1.52i)T + 67iT^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + (-0.527 + 0.527i)T - 73iT^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 + (1.08 + 1.08i)T + 83iT^{2} \)
89 \( 1 + 0.746T + 89T^{2} \)
97 \( 1 + (1 + i)T + 97iT^{2} \)
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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

−8.134742455777848600732805387095, −7.51894284228158324804447073234, −6.60422831080188628368382789234, −5.98471142495353414342854051223, −5.40570677644849095877746029554, −3.90598080644699329473097623558, −3.61549950742967848936014059394, −2.82616829718916339900585235393, −1.12859100393214578229272226660, −0.30363558197725891493714776578, 1.66272567367332962005250794462, 2.40668451642008303339672532144, 3.53111925359335885314747336777, 4.15654268613470361667224460598, 5.31823585457151582445623441982, 5.95484662604304644272696939506, 6.67839954944379568572636549513, 7.26060247340918086948321735461, 8.320662247462532104139613028212, 9.097963178579602245722023735945

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