Properties

Label 2-60e2-5.4-c1-0-41
Degree $2$
Conductor $3600$
Sign $-0.894 + 0.447i$
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  − 5i·7-s − 5i·13-s − 19-s + 7·31-s − 10i·37-s + 5i·43-s − 18·49-s − 13·61-s − 5i·67-s + 10i·73-s − 4·79-s − 25·91-s + 5i·97-s + 20i·103-s + 19·109-s + ⋯
L(s)  = 1  − 1.88i·7-s − 1.38i·13-s − 0.229·19-s + 1.25·31-s − 1.64i·37-s + 0.762i·43-s − 2.57·49-s − 1.66·61-s − 0.610i·67-s + 1.17i·73-s − 0.450·79-s − 2.62·91-s + 0.507i·97-s + 1.97i·103-s + 1.81·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 3600,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.281663470\)
\(L(\frac12)\) \(\approx\) \(1.281663470\)
\(L(\frac{3}{2})\) 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 \( 1 + 5iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 5iT - 97T^{2} \)
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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

−7.87112260524321714467039080242, −7.72155833311923291907485731016, −6.81531475745839131249806414152, −6.09750651855915398744012837930, −5.10729334517241851998332098234, −4.31029782917434790029497105093, −3.61871279566415536413523222071, −2.73224647575575663369456357834, −1.28177264963244280190096979497, −0.39229657431660952652605236117, 1.60010398080805915820508866624, 2.41260182677136107623503162583, 3.21389631703215382989902552813, 4.45670430796568630472806689582, 5.03745612959033359564318205548, 6.06265219306703355665865262601, 6.39302450206883075958968155699, 7.37925041713015986535745687257, 8.458977922491354144463185872546, 8.717333090204791611798453361056

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